Educational achievement in rural Montana high schools by John Wesley Kimble A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Applied Economics Montana State University © Copyright by John Wesley Kimble (1974) Abstract: This study attempts to look at Montana's educational system and its impact on rural and urban students. It has two major objectives: first, to analyze the effects of school size on student achievement and, second, to analyze the factors determining achievement and their variation in different size schools. In order to accomplish this, a sample of sophomore and senior students was taken and they were administered an achievement test. The data collected was analyzed in light of project objectives. The study concluded that school size was not a significant factor in student achievement and that the factors that contribute to student achievement vary for schools of different sizes.  STATEMENT OF PERMISSION TO COPY - In presenting this thesis in partial fulfillment of the require­ ments for an advanced degree at Montana State University, I agree that the Library shall make it freely available for inspection. .I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by my major professor, or, in his absence, by the Director of Libraries, It is understood that any copying or publication on this thesis for financial gain shall not be allowed without my written permission. Signature Date ft /f /y/y1 EDUCATIONAL ACHIEVEMENT IN RURAL MONTANA HIGH SCHOOLS by JOHN WESLEY KIMBLE A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE . in Applied Economics Approved: Chaim; Head/IMaj or Department Graduate Dean MONTANA STATE UNIVERSITY Bozeman, Montana December, 1974 ill ACKNOWLEDGEMENTS I owe so much to so many it would be impossible to list all who helped in this effort. Deserving of special note are Dr. Gail Cramer, my advisor. Dr. Verne House and Dr. Douglas Bishop, my committee members. A very special thanks to my wife, Carol.. TABLE OF CONTENTS Page V I T A ............ ............................ •............ .. ii ACKNOWLEDGEMENTS...................... ill LIST OF TABLES.............................................. vi ABSTRACT '..................................................... viii CHAPTER I: INTRODUCTION........ •........................... I CHAPTER II: REVIEW OF LITERATURE .......................... 9 The Coleman R e p o r t ...................................... 10 Summary of Other Research Studies ......................... 12 Summary of Achievement Tests Used . . . . . . . . . . . . 17 Conclusion................................... 18 CHAPTER III: ECONOMIC THEORY AND METHODOLOGY............ . . 19 The Economic Model.............................. 19 The Variables...................................... ' ... 25 The Student Sample.......................... 2.9 Standardizing the Test Score. ..................... 33 CHAPTER IV: DATA DESCRIPTION. . ............................... 35 Overview of Testing..................................... 35 Summary of Test Results for Seniors and Sophomores. . . . 36 Summary of Results by School S i z e ........................ 44 Comparing Scores to SAT Norms ............................. 47 The Independent Variables .............................. 50 Student's View of School............................ 52 Per Student Expenditures....................... j. . 52 School Size .......................... . . . . . . 53 Father's Educational Level .................... . . 53 TV, Phonograph, Telephone, Paper ......... . . . . . 53 Books and Magazines. . . .........................■ . 53 Autos in F a m i l y ................................ . 54 Own Room and C a r .................................... 54 C o l l e g e ........................................ . 54 School Activities. .................... . . . . . . 54 Hours- Worked on the Job. . . . .....................54 Grade Point Average. ................................. 5,5 iv VCHAPTER. V : RESULTS OF STATISTICAL ANALYSIS.................... 56 Regression Analysis for All Students ........... 59 Regression By School Size.......... '................. .. . 64 Summary................................ .. . . ...........75 CHAPTER VI: SUMMARY AND' CONCLUSIONS ............ .80 ■ Summary........................................ ■ ........ 80 Conclusions................................. 81 APPENDIX...................................................... 89 APPENDIX A ..........' . ................... '........... .. 120 APPENDIX B . . . . . .............................. 123 BIBLIOGRAPHY............................................. 127 vi LIST OF TABLES ’ Table Page III-I VARIABLES TO BE USED IN THE REGRESSION ANALYSIS . . . . 24 I II-2 . SCHOOL SIZE BREAKDOWNS FOR ANALYSIS PURPOSES . . . . . . 26 III- 3 SAMPLING BREAKDOWN OF MONTANA SCHOOLS BY S I Z E........ 30 HI-4 ACTUAL STUDENT SAMPLE FOR GRADES 10 AND 12 . . . . . . 32 IV- I MEAN STANDARD SCORES FOR SENIOR STUDENTS BY SCHOOL. . . 37 IV-2 MEAN STANDARD SCORES FOR SOPHOMORE STUDENTS BY SCHOOL . ' 38 IV-3 STANDARDIZED TEST MEANS RANKED FROM HIGHEST TO LOWEST FOR SENIOR STUDENTS SHOWING SCHOOL NUMBER ............ 40 IV-4 STANDARDIZED TEST MEANS RANKED FROM HIGHEST TO LOWEST FOR SOPHOMORES SHOWING SCHOOL N U M B E R ............ .. . 41 IV-5 SUMMARY TABLE FOR THE ANALYSIS OF VARIANCE TESTING MEAN DIFFERENCES OF STANDARDIZED TEST SCORES FOR SOPHOMORES. 42 IV-6 SUMMARY TABLE FOR THE ANALYSIS OF VARIANCE TESTING MEAN DIFFERENCES OF STANDARDIZED TEST SCORES FOR SENIORS . . 42 I V-7 STANDARDIZED TEST MEANS FOR SENIORS BY SCHOOL SIZE. . . 45 IV-8 STANDARDIZED TEST MEANS FOR SOPHOMORES BY SCHOOL SIZE . 45 IV-9 SUMMARY TABLE FOR THE ANALYSIS OF VARIANCE TESTING MEAN DIFFERENCES BY SCHOOL SIZE FOR SENIORS................ 46 IV-IO SUMMARY TABLE FOR THE ANALYSIS OF VARIANCE TESTING MEAN DIFFERENCES BY SCHOOL SIZE FOR SOPHOMORES . ........... 46 TV-11 AVERAGE RAW SCORES FOR SOPHOMORES AND SENIORS (ALL STUDENTS) . ...................................... 48 ■IV-12 PERCENTAGE OF STUDENTS BELOW THE FIFTH STANINE ON EACH TEST BY SIZE AND TOTAL................................ 48 vii IV- 13 INDEPENDENT VARIABLES USED IN THE STUDY . . . . . . . . 51 V- I DEPENDENT VARIABLES SUMMARY ALL STUDENTS.......... 57 V-2 DEPENDENT VARIABLE SUMMARY BY S I Z E .......... .. . . . 58 V-3 . TOTAL SCORE EQUATION FOR ALL STUDENTS WITH GPA AS AN . . INDEPENDENT VARIABLE................................ . 61 V -4 TOTAL SCORE EQUATION FOR ALL STUDENTS WITHOUT GPA AS AN INDEPENDENT VARIABLE............... ...... 63 V-5 TOTAL SCORE EQUATION FOR SCHOOL SIZE 0-117 STUDENTS . . 65 V-6 TOTAL SCORE EQUATION FOR SCHOOL SIZE 0-117 WITHOUT.GPA. 66 V -7 TOTAL SCORE EQUATION FOR SCHOOL SIZE 118-261 STUDENTS . 68 V-8 TOTAL SCORE EQUATION FOR SCHOOL SIZE 118-261 WITHOUT GPA 69 V-9 TOTAL SCORE EQUATION FOR SCHOOL SIZE 262-1,040 STUDENTS. 70 V-IO TOTAL SCORE EQUATION FOR SCHOOL SIZE 262-1,040 WITHOUT GPA . . . . '...................... .... 71 V-Il TOTAL SCORE EQUATION FOR SCHOOL SIZE OVER 1,040 . . . . 73 V -12 TOTAL SCORE EQUATION FOR SCHOOL SIZE 1,040 AND UP . ■ WITHOUT G P A ....................... ...... ........... 74 V -13 GPA CORRELATION COEFFICIENTS WITH THE DEPENDENT VARIABLES . . . . . .................................. 76 V-14 THE EQUATION WITH QPA AS THE DEPENDENT VARIABLE . . . . 76 viii ABSTRACT This study attempts to look at Montana's educational system and its impact, on rural and urban students. It has two major objectives: first, to analyze the effects of school size on student achievement and, second, to analyze the factors determining achievement and their variation in different size schools. In order to accomplish this, a sample of sophomore and senior students was taken and they were administered.an achievement test. The data collected was analyzed in light of project objectives. The study concluded that school size was not a significant factor in student achievement and that the factors that contribute to student achievement vary for schools of different sizes. Chapter I INTRODUCTION The purpose of this study is to analyze how the resources used by a school system relate to educational achievement. This research is especially relevant to Montana because of the problem caused both by the low population density and the rural nature of the state. The Census of Population defines rural residents as those who live in the open country or in communities of less than 2,500 people. Montana is a large state which encompasses 147,138 square miles or more than 94 million acres. According to the 1970 census, its popula­ tion numbers 694,409 people. The census lists 135 cities, of which .32 have a population of 2,500 or more and are considered urban. . Slightly more than 53 percent of the population live in these 32 communities. The other 323,733 people or 47 percent of the population are rural. In I960, 49 percent of the population was considered rural which shows a slight rural to urban migration. Figures which represent the education of the population show that the urban population is better schooled than the rural portion, espec­ ially at the post secondary level. The 1970 Montana census showed the median number of school years completed for urban persons 25 years and older to 12.4 years, 12.1 years for the rural nonfarm 2population, and 12.2 years for the rural farm population. Of the urban people, 13.6 percent have four years of college or more while only 6.4 percent of the rural farm population has four or more years of college. These figures compare favorably with the national medians. The national median of school years completed for persons over 25 are 12.2 years for the urban population and 11.1 years for the rural population. The percentage of persons with four years of college or more in the United States is 10.7 while nationally the rural percentage is 6.7 [Commerce, 1970]. The educational system has historically shortchanged rural people. Low levels of educational achievement by rural youth are indicative of the poor quality of education [Coleman, 1966]. While data indicates that rural youth are getting a better education than. their parents it still lags behind that of their urban counterparts. Rural students drop out of school at an earlier age and fewer rural students go to college. Those who do go on to school have a hard time competing with the urban student [U.S. Commission on Rural Poverty, 1967: 41-44]. The various components of an educational system which include teachers, buildings, facilities, curriculum, and programs are usually of less quality in rural districts than in urban schools.. Low teacher salaries in rural schools do not attract or retain the. better 3teachers. Poor facilities also contribute to the lack of better teachers. In general, small schools lack the equipment that the large urban schools.can afford. Furthermore, students who live in rural areas are hampered by geographic isolation and what is called "small town milieu" [Sweeney, 1971: 4-8]. These limitations may inhibit them from learning new behaviors for coping with urban living The Coleman Report (a recent study which hypothesizes that rural students are getting short-changed educationally) concluded that the low educational level of the parents and a combination of com­ munity factors place rural students at a disadvantage even when they enter school. The majority of Montana's high, schools, about 65 percent, have less than■180 students. These schools account for 20. percent of the total number of high school students in the state. Sixteen of Montana's largest urban high schools have approximately 50 percent of the student enrollment. In 1964 Montana's expenditures were.about $570 per pupil in average daily attendance. In comparison, New York spent $790 per pupil in average daily attendance for the same period. In 1972, Montana spent 52.18 cents of every tax dollar on education [Montana, 1972: I,1,V.1]. In Fiscal Year 70-71, 73 percent of school funds came from the local level, 21 percent from the state, and .6 percent from the U. S. government [Montana Schools, 1972:1.1]. These figures have not varied by more than 2 percent in the past ten years. 4This large tax burden carried by the local taxpayer can cause in­ equities in school quality. Because of the spillover of educational benefits to areas outside the local school district, emphasis has been placed on supporting school systems through a broader tax base. The problems of rural schools, and the costs of schooling, have made the issues of school efficiency, school quality, and educational quality of interest to nearly every citizen. In the last decade, rising costs and increasing taxes have aroused the public’s interest in the economics of schools. Accountability and efficiency have become key educational issues. The problems of education require much knowledge about the educational process and educational outcomes Economics is the science that deals with the allocation of scarce .resources among unlimited wants. Education is not a free good and it does have a price. The resources used in its production could be used to produce other goods. Education must be considered an economic good. It is the efficiency in combining resources and com­ paring the benefits derived from education to alternative resources that interest the economist. Essentially, educational outcomes consist of two major economic components. The investment component refers to future years of increased earning power. The consumption component includes the immediate utility and long-run satisfaction of education for the individual. These are personal outcomes since the benefits are accrued directly by the student and his family. The 5.amount a student or M.s parents'spend on . education is determined by M s estimated benefits from increased productivity and enjoyment. Thus, the student will spend an amount on education that will equate private marginal benefits' to"private'marginal costs. As an economic good education contributes to society as well as to those directly involved'. This spillover effect has two important implications. One is that the costs should be borne by those who benefit from education; another is that efficiency in resource allocation suggests that the amount of education provided will be less than optimal if the external benefits are hot considered. Making education socially efficient and equitable requires that the cost burden be adjusted. Expenditures must be extended beyond the point where private marginal costs equal private marginal benefits so that ultimately private marginal costs plus external marginal costs equal private marginal benefits plus external marginal benefits; also some . means of allocating the costs to the proper private arid public bene­ ficiaries must be considered. Educational costs can be further divided into private and public costs. Private or individual costs include indirect costs such.as earnings foregone while in school and direct costs such as tuition, books, and supplies. Public.cost includes the cost of building and operating schools. The .relationship that links'.educational achievement and educational 6resources can be analyzed by using the logical constructs of economics In this analysis, we can assume that the school functions somewhat like a firm in the sense that it tends to maximize output within its resource limitations. In order to accomplish this, it is necessary to consider an educational production function. Educational pro­ duction can be depicted as a functional relationship which illustrates the maximum amount of educational output that could be produced by each and every set of inputs. In order to achieve the greatest out­ put for a given budget restriction, the decision maker must determine the combination of resources that will maximize output within the budget constraint. This condition is satisfied by purchasing and utilizing each of the inputs in such a combination that the last dollar expended on each of the inputs yields the same effect on output The flow then from budget to output is as follows: dollar budgets are used to purchase school inputs in resource markets (which include, markets for personnel, equipment, and so on); these resources are combined by the school administrator in some fashion; finally, the relationship between input and output and how they are combined can be represented by a production function. This study will consider two important areas in Montana's educa­ tional system. First, school quality and how it varies according to .school size will be investigated. If the student in the larger, more urban schools is getting a better education than the rural student 7then a possible shift of funding priorities may be in order. Second, those factors that determine educational achievement will be inves­ tigated. ■ Determining factors in differently sized schools may well indicate where to place educational resources. The study then has two major objectives: 1) To analyze whether or not school size is important in determining student achievement, and 2) To analyze factors that affect student achievement and to determine whether or not these factors vary.in different size schools. The procedure followed for achieving these objectives is as follows: • I) A survey of the literature relevant to the subject area of the study to help give direction and purpose to the project, 2) Instruments were either identified or designed to collect the data, 3) A sample of schools was drawn and permission was obtained to perform the required testing and data collection, 4) Testing and data collection was accomplished by two data collectors who were hired to visit each school as scheduled. The time to collect the data took about three months. 5) The data was coded and verified and processed on the Sigma-7 computer at Montana State University to allow statistical interpreta­ tion of the data and a thorough examination of what had been collected. 8and 6) The analysis of the data and the results are presented in Chapters IV and V. • Chapter II REVIEW OF LITERATURE Since nearly a quarter of this nation's population is enrolled in schools, it is important to determine what effect the schools have on what a student learns. Many educators, laymen, and researchers have contended that schools make little difference [Coleman, 1966]. Their contention is that academic performance is dependent on social and economic conditions outside the school. The impact of such thought, if true, would mean a great deal of tax money is being inefficiently used. For many years, educators and the general public have endeavored to make education more efficient. Early studies were conducted for the most part by professional educators. The main idea of these studies was to use per pupil expenditures to measure school quality. Many determinants were used and they ranged from student performance measures to a measure of how well the administration adopted innovative procedures. A fairly consistent conclusion was reached: more money means more effective schools. These analyses provided a strong incentive to increase spending for better student performance, but one thing was lacking: a measurement of the student's capabilities upon entering school and the influence of extracurricular activities upon student achievement. More recent studies have emphasized the 10 importance of social environment and have discounted the effects of schooling. It seems that in order to assess achievement, adequate account must be taken of both the social and school services to which the student is exposed. In order to do this accurately, it must be known what the status of the student is upon entering school, upon completion of school, and then how much of his achievement is attributable to the school. The controls that would be necessary to complete such a study would make it impossible. " Nevertheless, many studies (attempting to measure this achievement) have been undertaken. The Coleman Report mentioned in Chapter I best illustrates this new line of inquiry. It, like others, tends to emphasize the importance of the socio-economic environment of the student in determining his performance. The Coleman Report The Coleman Report (1966) has been the most widely discussed of all the studies. It was carried out as the result of the Civil Rights Act of 1964, Section 402, by the National Center for Educational Statistics of the U.S. Office of Education. James Coleman of Johns Hopkins University was responsible for the design, administration and analysis of the study. It is probably the most extensive attempt made to assess the quality of American education. Approximately 660,000 students, their teachers, and their•schools were surveyed. Other 11 questions were also asked, including those on the diversity of the curricula and the qualifications of the administrators. The report reveals that several of those factors are positively correlated with the performance of the pupil. The most significant school service variable in determining student achievement was the verbal ability of the teacher. This might be construed as a proxy of the teachers' intelligence and thus their ability to motivate and communicate in a manner that makes the subject understandable. The report showed a strong relationship of socio-economic status to.student achievement and in fact stated that most achievement was related to socio-economic factors. The report also found that a pupil's achievement is strongly related to the educational backgrounds and aspirations of the other students in the school. Expenditures per student were not significantly related to school achievement. The Coleman Report generated considerable criticism. Criticism by James W. Guthrie in Do Teachers Make A Difference [1970:25] challenges the statistical methodology and claims that the measure­ ments utilized are inadequate. Bowles and Levin [Winter, 1968:3-24] criticize poor sample response. They suggest that a better measure­ ment of facilities with less aggregation of data might show a better relationship between expenditures and student performance; 12 Summary of Other Resedircll Studies The results of other studies cannot be summarized concisely partly because of the large variety of measurements used and partly due to their diversity. The remainder of this survey provides a brief description of a selection of these studies and the conclusions that were drawn by each one. A study by Kiesling [1967:356-357] sampled 97 high schools out of 1,400 in New York. The measure of output was an achievement test. The input variables used were: pupil intelligence, socio-economic attributes of the community, per student expenditures, school size, and school growth rate. Kiesling discovered that school size was negatively related to achievement if at all, and that high expendi­ ture districts do a poorer job of educating pupils from low socio­ economic backgrounds than do low expenditure districts. A study in 1962 by Street, Hamblin and Powell [261-266], tried to relate school size to achievement. The study was done in East Kentucky and used the Stanford Achievement Test as a measure of school output. They classified the schools by categories of 0 to 100 students, 100 to 300 students, and 300 students and over. The con­ clusion reached was that the student in the larger school was likely to out-perform the student in the small school. They did warn, however, that factors other than size could have been responsible 13 for their findings. In 1957, a study by Shelly 11957] attempted to correlate eight factors with the quality of 39 South Carolina secondary schools. The eight factors used were teacher salary, teacher certification, scope of educational program, school size, quality of administration, facilities, socio-economic status' of the community, and the amount of money spent for instruction per teacher. These factors also accounted for 69 percent of the variation in quality. The scope of the educational program and the quality of administration seemed the most important factors. Little relationship between socio-economic status and quality was revealed, possibly because most of the school funding came from the state. A study completed at the University of Arkansas.in 1962 [Tread­ way, 1962:513] tried to link school quality to student achievement. Eight of the 39 factors used were deemed significant to student achievement. They were size of school district, financial support, supervisory services, class size, teacher turnover, high school expenditures, dual education programs, and teacher qualifications. A similar study by Simpson [1961:3499] found that six key factors explained student achievement. They were school size, rate of growth, expenditures, effort and capacity, program, and socio-economic status. It was not immediately clear what other variables he examined in his Jresearch. 14 A study by John RIew [1966:280-287] reviewed some works that showed variable results.. In one study, he found that there was little or no relationship between quality and school size. Riew1s own study chose Wisconsin high schools and used information from the State Department of Public Instruction. He found that per pupil expenditures decline as enrollment rises to 700 pupils. Expenditures rose when schools reached an enrollment of 701-900 pupils and then fell after that point. This fiscal study excluded capital expendi­ tures. Riew concluded that a larger sample of schools should have been used to strengthen his results. A study by J. Alan Thomas. [1962] employed data from project TALENT at the University of Pittsburgh. Thomas used more than 20 variables to analyze student achievement. Even after home and community factors were taken into account, the variables that seemed most related to achievement were: teachers' salaries, teachers' experience, and the number of library books in the school. He used scores on 18 different achievement tests and his sample was composed of 206 high schools in communities of 2,500 to 250,000 in 46 states. Other variables he found significant in influencing test scores were the size of class the student was in and the number of days a student spent in school. Samuel Bowles [1969] in a report to the U.S. Office of Education presented the econometric problems involved in estimating educational 15 production functions. Bowles focused on the meaning of such a function, output measurement, initial measurement, and the dimensions of the learning environment. He used data from the University of Pittsburgh's project TALENT and the Coleman Report to estimate educatonal production functions. He found that teacher quality was an important ingredient in student achievement. Bowles also explored four major characteristics of the home which affect achievement. The first factor was the verbal interaction and communication with adults. Secondly, he assessed the quality of such interaction and communication, using family size and education of the parents as criteria. Thirdly, Bowles looked at what motivates achievement, by examining the parents' attitude toward education. Fourth, he analyzed the degree of oppor­ tunity a student has to explore the physical environment in the home and measured it both by how often reading material was used in the home and by parents' income. The school environment was also measured by the educational level or verbal efficiency of teachers, school policies, extracurricular activities, class size, community support of education, and school facilities. Bowles found that in no instance was per pupil expenditure significantly related to achievement but found that most of the factors that were purchased by such expenditures showed a strong relationship. The results of his study showed several important factors including that parents' income better explained achievement than did parents' education. 16 There have been studies that measure educational quality through methods other than the use of achievement tests. One by Welch appeared in the -American Economic Review [1966:379]. The basis of this study was that the quality of schooling was considered a principal contributor to productivity. The study excluded from its population those who had attended college. The sample population was rural farm males older than 25 years. The study compared the income of the schooled representative to the income of a representative with no schooling. The percentage of students who returned to school was estimated for each year. It appeared that teacher quality did enhance school productivity. The validity of this study might be questioned because that part of the population that was rural schooled but was forced to move to the city for employment was excluded. The study also aggregated data on a statewide basis, and less aggregation might have provided more explicit data. The concept of investment in human capital set forth in the study by Welch has also been investigated by Theodore W. Schultz [1971]. The major economic benefit of education is the increased productivity of the person receiving it. The assumption that the wage a person is paid is equal to his contribution to output (marginal product) is important to this theory. It follows then.that the increased pro­ ductivity derived from investing in educational will have a positive effect on earnings. Schultz's studies conclude that education is an 17 important factor in influencing economic growth. The discrepancy between the growth rates of national income and national resources leave an unexplained area that might, be explained in part by the increased investment in human capital. Summary of Achievement Tests Used Bowles [1969] chose three measures of output: reading compre­ hension, mathematics competence, and a composite score based on reasoning, creativity, vocabulary, and English. Each achievement test measured different levels of achievement. The Coleman Report [1966] used a test series designed and administered by Educational Testing Service, Princeton, New Jersey. Other studies used a variety of tests ranging from unspecified aptitude and achievement tests to specific tests. Some studies used complete batteries of achievement tests, others only an individual test. Those studies that were based on data collected by.the Coleman Report or by project TALENT had specific tests. Street, Powell, and Hamblin [1962:261-266], in their study in Kentucky, used the Stanford Achievement Test. Nearly as many different achievement tests as studies were used. A few of the studies reviewed used whatever scores were available and then converted them to a standard norm for analysis. Administering a specific test is costly and time consuming but does allow for data control and for more convenient collection of either data on the social background of 18 the student. The Seventh Mental Measurement Yearbook [Buros, 1972] lists 36 achievement batteries, plus various tests in specific areas. The choice of test or tests would depend on the areas to be tested. Measuring achievement is by no means an exact science, but there are at least six good general achievement tests that should be adequate. Conclusion Ten studies and their research techniques have been reviewed in this chapter. Nine of these used achievement tests as a measure of school output. Two looked at the productivity of the subject in determining the effects of schools. School size was a variable in seven of the ten studies and was not significant in four, possible significant in one, and significant in only two cases. The variable that seemed consistently significant in the studies that used it was socio-economic status. In no instance did per student expenditures show a positive relationship to achievement, yet the items that the money bought, such as facilities, teachers, and extracurricular activities, did show a relationship in some of the studies. Each study used some unique var­ iable that showed a significant relationship to achievement. Chapter III ECONOMIC THEORY AND METHODOLOGY The Economic Model Production is the process of combining inputs to create outputs of a specified form. An input is simply anything which a firm buys for use in its production or other processes and an output is any commodity which a firm produces or processes for sale. Production is assumed to occur only if the outputs are of more value to society than the inputs. This relationship between the input and output variables is often referred to as a production function. The educational production function relates school and student inputs to a measure of school output. Representation of the educational production, process in this form is of interest in studies of human capital formation as.well as studies in optimum resource allocation in education. There are certain complexities in applying a production function to education. The most complex of these is the student. Each student is an individual with traits and characteristics not homogeneous to the group. The teacher, laws and regulations, and the community's educational concept add to the complexity of the situation. The production function is based on the idea that output depends on the inputs used and that there is a unique output for each possible 20 combination of inputs. Symbolically, the production function can be represented by: Y = f (X) where output Y is dependent on the amount of input X used given the existing technology. In order to present a more accurate picture of the model, more than a single input is necessary. The production function would then take the form Y = f(X11X21X3, ..., Xn) where Y again represents the output and the inputs are represented by X11X31X^1 ..., X^. The output can be varied by using different amounts of X1, X31 etc. or by using a different quantity of one input and holding the others constant. This can also be stated in terms of obtaining a fixed output Y and by looking at the different possible combinations of the input variables that it would take to obtain a fixed output of Y [Doll, 1968: 39-60]. The optimum combination of the input factors can be determined by calculation of the marginal product of the input variables. The marginal product of an input factor can be defined as the addition to output of the last unit of the input factor added. If we assume that the primary objective of a firm is to produce as efficiently as possible then the cost outlay should be as low as possible for the determined level of output. In order to accomplish this the marginal 21 product of a dollar's worth of one output must equal the marginal product of a dollar’s worth of every other input used. The model in this particular application involves one dependent variable and several independent variables. The basic production function will take the form: Al = f^ + + ... + + Ui where: = the achievement score (Dependent Variables); fg through fn = the parameters of the production function; Xn^ = the amount of input n devoted to observation of student i's education and n - I through n (Independent Variables); and = a disturbance term. The school input factors are dependent on a system of simul­ taneous equations representing the school administrator's social welfare function, the budget constraints, and the educational pro­ duction function. Because of this, any estimation.of the parameters in this model will lead to inconsistencies. One way around this problem is to assume that school administrators probably do not select school inputs as if they were maximizing a well-defined production function. This assumption has some basis in actual practice as the administrator lacks perfect knowledge and is subject to political . and legal constraints. This assumption causes another problem in that 22 if the school administrators do not conform to any systematic optim­ izing model, then the observations of the data are not technically efficient. Thus we get some sort of an average production function [Bowles, 1969: 10]. The knowledge available on learning relationships makes the specification of an educational production function difficult at best. The concept of the margin and diminishing return is not well established in the industry. Therefore, a linear function would seem somewhat superior for. the purposes of this study. Possible positive interactions of our inputs would be another reason for using a linear form.. The restrictions for a linear form are many and severe, but for reasons of simplicity, the linear additive form presented above will be used. This study focuses primarily on the economic consequences of schooling. Ideally, the output measures would include income and social, adjustments of the individual after schooling. Lacking the > ' opportunity to measure post school adjustments, a proxy, in the form of an achievement test score, will be used. Policy making is primarily concerned with the parameters of the production function and the marginal products of the inputs for movement toward optimal input combinations. It is doubtful that the marginal product of the same input for different groups of students can be compared. 23 The score on an achievement test is an ordinal measurement. There is no zero point and no well-defined unit of measurement. This has implications in the area of the marginal rate of substitution. Although the marginal rate of substitution is valid theoretically, the absoluteness of the measure of marginal product is not. Among students scoring at different parts of the measurement scale, equal units of increase in scores are not comparable [Gardner, 1965: 24]. Some assumptions must be made before we can begin this analysis. First we must assume that the variables used represent quantities observed without error. Although the data are subject to some degree of error in measurement, it seems prudent to consider these errors negligible in light of the unobservable elements (the Ihl terms). The error term (U^l) must be assumed to be normally distributed with mean zero and variance sigma squared [Malinvaud, 1966: 172-175]. Another of the purposes of this study is to attempt to determine those factors that affect student achievement. After all the data are collected, a linear regression program [Nie, 1970: 174] will be used to obtain the necessary statistics. Four dependent variables and 18 independent variables will appear in the model. Table III-I shows the variables to be used in the regression. A more definitive description of each variable will be included in Chapter IV. For purposes of analysis, the sample will be examined in two 24 TABLE III-I. VARIABLE'S TO BE USED IN THE 'REGRESSION ANALYSIS. Variable Description Dependent Variables: English test score - Standardized Numberical competency score - Standardized Reading test score - Standardized Total test score Independent Variables: Student's view of school Average daily preparation per teacher Per student.expenditure Student-teacher ratio Beginning teacher salary Year school facility was built School size Father’s highest grade level attained Items in the home (TV, radio, phonograph, paper) Number of yearly books and monthly magazines Number of automobiles in the family Student's own room and car Student's college plans Community size Student's involvement in school activities Hours student works outside school Amount of travel the student has done High school grade point average 25 different ways. First, each, of the 27 schools will be placed in I of 4 categories, broken down according to student size. A description of these categories is shown in Table III-2 and the number.of schools in each is indicated. This will isolate the effects the variables have on achievement in schools of a different size. Second, all students will be analyzed collectively. As mentioned, the production process of schools is complex. About the most that can be expected from estimation is a discovery of some of the relationships of the educational process. The Variables The intent of this study is to examine the effects of variables representing student socio-economic status and the school and their relationship to selected dependent variables represented by achieve­ ment test scores. Three sections, reading, English, numerical com­ petency, of the Stanford Achievement Test, High School Battery will be used as dependent variables. There are several reasons for choosing a less extensive testing regimen: I) The cost of administering the complete 6-hour battery would be prohibitive; 2) remaining sections of the test may have favored schools with broader curriculums; and 3) most cooperating schools would be disrupted less by an abbreviated testing program. The complete test battery was given in any school TABLE HI-2. SCHOOL SIZE BREAKDOWNS FOR ANALYSIS PURPOSES. Student No. of Size Per School Schools 1 0 to 117 7 2 118 to 261 7 . 3 262 .to 1,040 7 4 1,041 and Larger _6 TOTAL SCHOOLS 27 26 27 that requested it. The SAT Battery was chosen based on Buros 11972]. This review of the tests rates it of high quality overall. Both the numerical competency and reading tests ranked high. The English test was not well ranked but rated high enough for use in this study. A number of independent variables ware collected for the study. Since the projected model needed proxy values' for student socio­ economic status and school influence factors, the following informa­ tion was collected by questionnaire for each twelfth grade student (see the appendix for a copy of the questionnaire):■ 1. Esther's occupation; 2. Esther's highest educational level; 3. Mother's occupation; 4. Mother's highest educational level; 5. Major provider; 6. Family size; 7. Number of cars in family; 8. Whether they have television, telephone, newspaper, and radio: 9. Number of magazine subscriptions; 10. Number of books purchased per year; 11. Whether the student has their own room, and car; Location of residence;12. 28 13. Part-time work; 14. Hours per week spent on studies outside school; 15. Travel in Montana, U.S., world; 16. School activities; 17. Favorite subject; 18. After school plans; 19. Time in community; 20. High school grade point average. The data used to represent school inputs were collected at the school site and from the trustees1 report to the Superintendent of Public Instruction. The expenditures and revenues for each school district were also collected from the State Superintendent’s Office. In multi-school districts, the chief accounting officer of the dis­ trict aided in determining the funds allocated to each school. Other school data included information on teachers and school facilities. Data on the teachers in the system included degree level, experience, and daily preparations. Data on the school facilities including building age, number of classrooms, full-time non-teaching staff, starting salary level for teachers, students bussed daily, and com­ munity population was gathered. All of these data were added to the information on each student. Data was also collected on student attitudes. One of the questions asked, for example, was ..whether the 29 student liked school. Another question asked the student was who had influenced their plans greatest after schooling.was completed. The Student Sample The process below describes how the schools and the individual students were chosen. All public high schools in Montana were listed from largest to smallest according to size of the school. Statistics [Montana, 1972] from the Office of the Superintendent of Public Instruction for 1972 showed 47,045 high school students in 167 operating high schools. These students were then broken into 10 groups with 10 percent of the total students in each. Then the list of schools was broken into 10 groups each containing about 10 percent of the students. Table III-3 shows the final groups and the sizes of the schools included. Five percent of all high school students were to be tested. The method of selection varied because of the number of schools in each of the size categories. Schools that fell in the first three size categories were chosen at random until the approximate predetermined number of sample students' was obtained. In size cate­ gories four, five, six, seven, and eight, not only were the sample schools chosen at random but in most cases the sample of students within schools were picked at random. In the largest categories, a sample of students was taken from all the schools to obtain the 30 TABLE III-3. SAMPLING BREAKDOWN OF MONTANA SCHOOLS BY SIZE. Size No. of Schools No. of Students Approx. No. of Students to Test No. of Schools % of School to test I. 0- 117 74 4,952 260 7 100 2. 119- 180 33 4,867 270 4 100 3. 181- 261 23 4,890 340 ■ 3 100 4. 272- 470 12 4,373 300 3 50 5. 473- 621 9 4,694 260 3 30 6. 639-1,040 6 4,748 260 2 30 7. 1,207-1,677 3 4,452 260 2 16 8. 1,815-1,905 3 5,575 240 2 9 9. 1,936-2,271 2 3,976 300 2 12 10. 2,247-2,271 2 4,518 330 3 11 167 47,045 2,820 30 31 desired number of students. This procedure left 30 schools. Three of the 30 schools would not participate in this survey. The size of these schools placed them in a category which allowed no substitutes so 27 schools were used in the project. Testing dates were set with each of the schools. Table III-4 shows a breakdown of students tested. Discrepancies appear because in some cases the total population, rather than a sample, were tested upon request. Absences and limitations set by the school administra­ tors also contributed to the discrepancies. Other schools asked that the total school population (grades nine and eleven) be.tested at the same time as the tenth and twelfth grades. The figures of the ninth and eleventh grades tested were not included in Table III-4. Table III-4 shows the total number tested in the school as well as the breakdown for the sophomores (grade 10) and the seniors (grade 12). . The protection of student anonymity was a slight problem during, data collection. In order to return the test results to the schools, the project agreed to supply to each school a label with student identification and the test scores. Each test answer sheet was pre­ numbered before the project began. The respondents in grade twelve transferred this number to the questionnaire and it was the only item entered into the project data files as a means of student identification. 32 TABLE III-4. ACTUAL STUDENT SAMPLE FOR GRADES 10 AND 12. Size Total No. Tested Grade 10 Tested Grade 12 Tested I. 0- 117 356 • 132 124 2. 119- 180 193 127 . 66 3. 181- 261 237 138 99 4. 272- 470 548 348 200 5. 473- 621a 182 104 78 6. 639-1 ,040 240 132 108 7. 1,207-1 ,677 197 128 69 8. 1,815-1 ,905 164 97 67 . 9. 1,936-2 ,040 169 105 64 10. 2,247-2 ,271% TOTALS 2,186 . 1,311 875 a One school declined to participate. hD Both schools declined to participate. 33 The method of selecting a sample of students in individual schools varied. In some cases, particular classes thought to be representative of the students were chosen. In some schools, the sample was taken from alphabetical lists or by random selection from another file source. Two schools utilized a computer to draw their sample. One school agreed to participate in the project on the condition that student participation be on a volunteer basis. In this instance, project personnel met with the students, explained the project and asked for their participation. Another school agreed to participate only.on the condition that school personnel administer the test. In addition, a small part of the student data had to be elim­ inated becasue of responses that did not make sense. These responses numbered less than 10. Standardizing the Test Score Each student was given three tests from the Stanford Achievement Test, High School Batterv. The result was a raw score for each student and each test. The raw score alone does not have much meaning and must usually be related to other scores achieved. In order to make this kind of comparison, the scores must be translated to means and standard deviations of a standard value. The conversion formula used for this process was: 34 Standard Score - 5 0 + 1 0 raw score - mean raw score standard deviation This resulted in a standard score with a mean of 50 and a standard deviation of 10 [Roscoe, 1969: 53-57]. The mean raw score and the standard deviation were obtained from the Stanford Achievement Test Manual [Gardner, 1965: 13-14]. The specific formulas used for each test for each senior tested were as follows: raw score - 41.8standard reading score - 50 + 10 [ 11.18 ] standard numerical competency score - 5 0 + 10 [ raw. sc°rpQ— ^ ^ ] . 8.88 standard English score - 50 + 10 [ ] . The specific formulas used in the case of the sophomores tested were as follows: standard reading score -5 0 + 10 [ r— - ^l^gg— ] standard numerical competency score - 5 0 + 10 [ raw- standard English score -5 0 + 10 [ r-— sc°^e^ 5_ ^^ The score transformation does not change the shape of the distri­ bution of the raw scores, it changes only the mean and standard devia­ tion. The scores received on tests from different subject areas are not comparable and cannot be combined. Standard scores have the same mean and standard deviation and can usually be combined without objection. Chapter IV DATA DESCRIPTION The results of the data collection and a discussion of the depend ent and independent variables used in the regression analysis are contained in this chapter. The results of the test scores for the sophomores and seniors are presented and then analyzed separately. Each of the independent variables are briefly described. Overview of Testing The achievement test used in the study was the Stanford Achieve­ ment Test (SAT), High School Battery. Three of the tests were used— English, numerical competency, and reading. Two. individuals were hired .to administer the tests. One was certified to teach mathematics at the secondary school level but had no teaching experience. The second was a certified elementary teacher with eight years teaching experience and a.master's degree in education. The tests were admin­ istered according to the instructions provided by the test company and were scored by the Test Scoring Service Department at Montana State University. s The tests are designed to measure the educational achievement of students in school. It takes 40 minutes' to administer each of the three tests. The English test consisted of three parts; the numerical 36 competence and reading tests were a single section each. In order to establish norms and a Basis for standardization, the test was admin­ istered to 22,699 students- on a regional Basis. Tor each student the raw scores must Be converted to a standard score which can in turn Be translated to either a percentile rank or stanine or Both. The Stanford Achievement. Test people provide the necessary information for this pro­ cess of standardization. The standard score they provide has a median of 50 and a standard deviation of 10 with a normal distribution. This standardization process is different for each grade, or class. Per­ centile rank allows the studentTs performance to Be compared with the norm group. For example, a percentile score of 60 would mean the student was equal to or greater than 60 percent of the other students in his group. The stanine is a value which is represented on a simple nine-point scale of normalized scores. The scores range from a low of I to a high of 9 with the value 5 always representing the average performance for students in the norm group. Specific information on the tests may be obtained from the SAT manual [Gardner, 1965]. Summary of Test Results'for Seniors and Sophomores Tables IV-I and IV-2 show by school the mean scores which sopho^ mores and seniors achieved for each test. The scores are standardized By the process presented in Chapter III of this study and not according to. the description in the SAT manual. The means are listed in four 37 TyNBLE IV-I. 5 I ErorEE FOE SENIOR STUDENTS BV SCU00L. School English Reading Numerical Total Number Mean Mean Mean Mean I 46.39 44.56 46.60 137.56 2 49.96 50.47 52.77 153.21 3 56.55 55.21 56.11 167.88 4 49.64 48.50 50.33 148.48 5 40.37 50.35 49.47 150.20 f. 52.78 51.40 57.89 162.08 7 45.53 42.00 45.43 132.97 8 55.63 52.35 52.13 160.11 Q 57.83 57.63 56.90 172.37 10 44.57 44.07 44.54 133.18 11 56.76 57.37 53.66 167.80 12 50.81 53.39 53.76 157.56 13 46.25 48.80 47.10 142.15 14 50.71 48.10 47.61 146.43 15 50.73 50.51 51.78 153.03 16 48.29 45.33 47.83 141.46 17 51.46 51.46 53.07 156.01 18 54.28 55.18 57.88 167.35 19 46.15 47.28 49.67 143.11 20 48.25 ?o.l2 52.02 149.40 21 48.76 51.78 51.24 151.78 22 46.93 47.36 46.85 141.14 23 52.87 54.97 52.°4 160.79 24 51.16 51.12 48.60 150.99 25 48.80 40.75 40.77 148.32 26 53.33 54.04 54.24 161.62 27 48.65 48.60 40.60 146.86 38 TABLE IV-2. MEAN STANDARD SCORES FOR SOPHOMORE STUDENTS BY SCHOOL. School Number English Mean Reading Mean Numerical Mean Total Mean I 46.99 48.04 53.61 148.6 2 47.53 47.19 49.85 144.6 3 54.37 55.99 55.84 166.2 4 51.69 51.67 54.15 157.5 5 47.64 47.99 48.18 143.8 6 54.84 51.06 53.70 159.6 7 48.36 46.17 49.31 143.8 8 50.60 46.10 48.63 145.3 9 50.40 49.78 50.79 151.0 10 48.89 46.33 50.31 145.5 11 51.46 53.93 54.48 159.9 12 52.52 53.22 55.35 161.1 13 47.66 49.62 50.25 147.5 14 55.13 52.96 54.74 162.6 15 47.71 48.10 47.84 143.7 16 51.68 49.79 50.48 151.9 17 51.69 51.86 52.04 155.6 18 54.81 54.40 56.42 165.6 19 49.66 50.72 51.22 151.6 20 48.55 46.82 50.16 145.5 21 51.79 49.95 52.03 153.8 22 53.14 51.73 50.85 155.7 23 55.61 56.26 55.79 167.7 24 52.29 52.51 51.19 156.0 25 52.87 50.28 51.82 155.0 26 57.24 57.73 57.02 172.0 27 49.33 50.77 50.15 150.2 columns by school number. The four columns- represent the three subject areas which were tested and a total mean, which is a composite of the first three means. The same scores are ranked from high to low in tables TV-3 and TV-4. According to the data, the schools perform fairly consistently across the three subject areas. The schools are numbered from I to 27 according to their size (I is small and 27 is large), and the pattern in these tables does not indicate that the largest school achieved the highest means or that the lowest means was achieved by the smallest school. After the raw test scores were converted to standardized scores, the means were compared for each test. Tables IV-5 and TV-6 show the . results of an analysis of variance procedure which was used to test the hypothesis of equal means. Each table includes the overall mean for the test, the calculated F-value, and the within mean square, for both seniors and sophomores. As shown oh the table, the F-value in each case has a significance level of .01 which indicates that the hypothesis that all means are equal does not hold true here. The results were further analyzed to determine the differences among individual means. The method used was the Scheffe—test for all possible comparisons [Roscoe, 1969: 238-242]. The Scheffe formula appears below: 40 TABLE IV-3. STANDARDIZED TEST MEANS RANKED FROM HIGHEST TO LOI7FST FOR SENIOR STUDENTS SHOWING SCHOOL NUMBER. English Reading Num. Comp. Total Mean School # Mean School # Mean School # Mean School # 57.837 9 57.633 9 57.895 6 172.376 9 56.762 11 57.375 11 57.883 18 167.888 3 46.556 3 55.216 3 56.907 9 167.802 11 55.631 8 55.188 18 56.177 3 167.351 18 54.281 18 54.976 23 54.243 26 162.086 6 53.339 26 54.045 26 53.665 11 161.627 26 52.871 23 53.393 12 53.364 12 160.796 23 52.788 6 52.351 8 53.079 17 160.116 8 51.467 17 51.782 21 52.948 23 157.569 12 51.169 24 51.469 17 52.778 2 155.014 17 50.811 12 51.403 6 52.134 8 153.216 2 50.730 15 51.129 24 52.072 20 153.030 15 50.712 14 50.517 15 51.783 15 151.784 21 50.378 5 50.477 2 51.241 21 150.903 24 49.961 2 50.351 5 50.338 4 150.200 5 49.649 4 49.755 25 49.775 25 149.406 20 48.800 25 49.122 20 49.670 19 148.489 4 48.762 21 48.809 13 49.604 27 148.329 25 48.650 27 48.607 27 49.471 5 146.860 27 48.299 16 48.502 4 48.695 24 146.439 14 48.258 20 48.109 14 47.834 16 143.113 19 46.932 22 47.362 22 47.619 14 142.159 13 46.395 I 47.287 19 47.100 13 141.468 16 46.250 13 45.335 16 46.852 22 141.145 22 46.157 19 44.568 I 46.601 I 137.565 I 45.539 7 33.075 10 45.431 7 133.18Q 10 44.572 10 42.001 7 44.542 10 132.971 7 41 TABLE IV-4. STANDARDIZED TEST MEANS RANKED FROM HIGHEST TO LOWEST FOR SOPHOMORES SHOWING SCHOOL NUMBER. English Reading Num. Comp. Total Mean School # Mean School Zf Mean School # Mean School # 57.24 26 57.73 26 57.02 26 172.0 26 55.61 23 56.26 23 56.42 18 167.7 23 55.13 13 55.99 3 55.84 3 166.2 3 54.84 6 54.40 18 55.79 23 165.6 18 54.81 18 53.93 11 55.35 12 162.6 14 54.37 3 53.22 12 54.74 14 161.1 12 53.14 22 52.96 14 54.48 11 159.9 11 52.87 25 52.51 24 54.15 4 159.6 6 52.52 12 51.86 17 53.70 6 157.5 4 52.29 24 51.73 22 53.61 I 156.0 24 51.79 21 51.67 4 52.04 17 155.7 22 51.69 4 51.06 6 52.03 21 155.6 17 51.69 17 50.77 27 51.82 25 155.0 25 51.68 16 50.72 19 51.22 19 153.8 21 51.46 11 50.28 25 51.19 24 151.9 16 50.60 8 49.95 21 50.85 22 151.6 19 50.40 9 49.79 16 50.79 9 151.0 9 49.66 19 49.78 9 50.48 16 150.2 27 49.33 27 49.62 13 50.31 10 148.6 I 48.89 10 48.10 15 50.25 13 147.5 13 48.55 20 48.04 I 50.16 20 145.5 10 48.36 7 47.99 5 50.15 27 145.5 20 47.71 15 47.19 2 49.85 2 145.3 8 47.66 13 46.82 20 49.31 7 144.6 2 47.64 5 46.33 10 48.63 8 143.8 5 47.53 2 46.17 7 48.18 5 143.8 7 46.99 I 46.10 8 47.84 15 143.7 15 42 TABLE 1V-5. SUMMARY TABLE FOR THE ANALYSIS OF VARIANCE TESTING MEAN DIFFERENCES OF STANDARDIZED TEST SCORES FOR SOPHOMORES. Mean E-Value Within M.S. English 51.21 4.94184 77.4907 Reading 50.82 4.73768 91.7132 Numerical Competency 51.65 4.51740 80.3804 TOTAL SCORE 153.7 5.49332 482.778 TABLE IV-6 SUMMARY TABLE FOR THE ANALYSIS OF VARIANCE TESTING MEAN DIFFERENCES OF STANDARDIZED TEST SCORES FOR SENIORS. Mean F-Value Within M.S. English 50.15 3.42452 81.6572 Reading 50.23 3.47331 96.1116 Numerical Competency 50.95 4.03208 80.5644 TOTAL SCORE 151.3 4.0661 602.563 43 . The components of this formula are broken down as follows: X = mean of group n; Sw = within' mean square; . Nn = number in group n; and k - number of groups. After being calculated, this F-value is then compared to the tabled F-value with (k-1) and I (Nn)-k] degrees of freedom. The results indicate that there was no significant difference among the individual means [Roscoe, 1969: 238-242]. This contradicts the results of the analysis of variance test for significance. The analysis of variance test is affected by non-normality and heterogeneity of variances when sample sizes differ. The Scheffe' method is considered very conservative and more rigorous than other multiple comparison methods [Snedecor, 1967: 278], and it is also quite insensitive to departures from normality and homogeneity of variance. When all of the above facts are considered, the possibility of the analysis of variance method, showing different results than the Scheffe' method, becomes quite apparent [Ferguson, 1966: 269-297]. The analysis of variance does provide a more powerful test and it is reasonable to conclude that there, is some significant . difference between the largest and smallest means. There may be other differences, but these cannot be proven with the methods used. 44 Summary of Results by School Size Tables IV-7 and IV-8 show the mean scores for each school as catagorlzed by size. As described in Chapter III, the 27 .schools in which the achievement tests were administered were broken into 4 categories according to the number of students attending. Table III-2 shows this breakdown and the number of schools represented in each classification. Tables IV-7 and IV-8 show the mean score that the seniors and sophomores obtained on each test. It also lists the composite mean and the number of schools that, were included in each size classification. The results of an analysis of variance test among the mean differences for the seniors appears in table IV-9 by test area. Table IV-IO contains the same information for the sopho­ mores . The means shown in Table IV-7 are very closely grouped. The results of the analysis of variance run with a calculated F-value indicate that the F-value for the numerical competency test was significant at the 0.1 level (Table IV-9). The Scheffe1 test of multiple comparisons showed no significant difference between the largest and smallest means. No difference is apparent in the achievement levels of seniors from different size schools. The seven smallest schools had the highest mean of all the schools in the numerical competency test and in the total mean score. The six largest schools. 45 TABLE IV-7. STANDARDIZED TEST MEANS FOR SENIORS BY SCHOOL SIZE. Size School Numbers Included English Mean Reading Mean Numerical Mean Total' Mean I I through 7 50.87 49.93 51.95 152.7 2 8 through 14 50.93 50.98 50.03 151.9 3 15 through. 21 ' 49.63 49.83 51.60 151.1 4 22 through 27 50.06 ■ 50.57 49.84 150.5 TABLE IV-8. STANDARDIZED TEST MEANS FOR SOPHOMORES BY SCHOOL SIZE.. Size School Numbers Included English Mean Reading Mean Numerical Mean Total Mean I I through 7 50.20 49.73 52.09 152.01 2 8 through 14 50.95 50.24 52.08 153.27 3 15 through 21 50.84 50.23 51.45 152.53 4 22 through 27 53.41 . 53.21 53.80 159.43 46 TABLE IV-9. SUMMABY TABLE FOR THE ANALYSTS OF VARIANCE TESTING ME,AM DIFFERENCES BY SCHOOL SIZE FOR SENIORS. Mean F-Value English 50.15 1.02760 Reading 50.23 0.603653 Numerical Competency ' 50.95 2.55672 TOTAL SCORE 151.3 0.244799 TABLE IV-IO. SUMMARY TABLE FOR THE ANALYSIS OF VARIANCE TESTING MEAN DIFFERENCES BY SCHOOL SIZE FOR SOPHOMORES. Mean F-Value English 51.21 1.604833 Reading 50.82 1.83860 Numerical Competency 51.65 0.62460 TOTAL SCORE 153.7 2.47970 47 size class four, had the lowest total score. Size class two had the high mean score in English and reading while size class three ranked lowest in these two areas. These results also show that the mean scores in the numerical competency area differ. Other differences are not evident. Table IV-8 shows' that in size class four, sophomores in the six largest schools scored highest in all of the three areas tested and in total score. The seven smallest schools ranked low in English and reading and class three schools scored low in numerical competency. Analysis of variance revealed a significant F-value only in the total score (Table IV-10); however, the Scheffe' test showed no significant difference in this case. Again, data suggests that there is some significant difference between the largest total score mean and the smallest total score mean, but that no other valid=, conclusions can be drawn. Comparing Scores to SAT Norms Tables IV-Il and IV-12 compare the mean scores of the students to the percentile and stanine rankings provided by the Stanford Achievement Test (SAT). Table IV-Il shows the raw mean in each test converted to the SAT standard score, which can be placed in stanine and percentile rankings. All of the mean scores are in the fifth stanine which indicates average performance. Whereas the mean achieved 48 TABLE IV-Il. AVERAGE RAW SCORES FOR SOPHOMORES AND SENIORS (ALL STUDENTS). ________SAT Conversion________ Raw Mean Standard Stanine Percentile Sophomore English 53.346 49 5 50 Numerical Compentency 28.229 50 5 52 Reading 36.633 50 5 50 Senior English 59.707 ' 53 5 44 Numerical Compentency 31.046 • 53 5 48 Reading 42.1059 54 5 54 TABLE IV-12. PERCENTAGE OF STUDENTS BELOW THE FIFTH STANINE ON EACH TEST BY SIZE AND TOTAL. Students Size Class I 2 3 4 ai: ------------------ Percent English Sophomores 23 23 27 12 21 Seniors 33 38 . 43 38 39. Numerical Competency Sophomores 19 20 25 17 2.1 Seniors 37 42 39 47 41 Reading Sophomores 31 28 29 ■ 16 25 Seniors 54 49 47 44 48 49 by the sophomores placed them equal to or greater than 50 percent of other students in their group in each test area, the seniors ranked at the forty-fourth percentile in English, the forty-eighth percentile in numerical competency, and fifty-fourth percentile in reading. This is not quite as high as the sophomores. Table:',IV-12 shows the percentage of sophomores and seniors whose scores fell below the fifth stanine. It is broken down according to subject area and grade level. The smallest schools ranked highest in English and numerical competency and lowest in reading for seniors. Class three schools ranked worst in English and class four schools performed lowest in numerical competency and highest in reading for seniors. The sophomores ranked fairly well in all areas, as the largest size class ranked best in all tests. The smallest schools ranked lowest in reading and class three schools ranked lowest in English and numer­ ical competency. In every case, the sophomores outscored the seniors. A couple of contributing factors are I) that the material in the test was basic and the sophomores were probably closer to actual course work in each area, and 2) that the students in the last two years of high school have a tendency to take course work somewhat removed from the basic reading, English and numerical competency courses. The sophomores in the large school systems do seem to rank better than those in the other three size classifications, but at the senior level there doesn't 50 seem to be a significant difference in the ranking by school size. The Independent Variables The study group collected a large number of variables for possible inclusion into the regression analysis. From this pool, the final 12 variables used were chosen on a basis of the study objectives and a review of other studies. This portion of the report will cover the variables collected but not used in the study first and then will give a brief description of those variables chosen. Variables were collected from three sources, the student, the school, and the State Superintendent of Public Instruction. The var­ iables were examined in light of their relationship to the student, the school, and to one another. On this basis, the final list of 12 variables representing the student, the socio-economic status of the student and the school were picked. Table IV-13 shows each selected variable accompanied by the mean and standard deviation for the seniors tested. The occupation of the student's mother and father was one variable rejected for use in the study. In order to make this data usable, a conversion process had to be used to convert the occupation to a factor representing socio-economic status. The conversion process used was based upon a list of occupations published by the National Opinion Research Center in a study by North and Hatt. Each occupation was 51 TABLE IV-13. INDEPENDENT VARIABLES USED IN THE STUDY. Variable Description Mean Standard Deviation Student view of school .6594 .4742 Per student expenditures 3045.0837 919.4023 School size 615.3920 610.9244 Father's educational level 11.3326 4.1516 TV, phonograph, phone, paper .9689 .0999 Books and magazines 12.9154 7.5022 Autos in family 2.8971 1.4105 Own room and car .6423 .3332 College .4965 .4389 School activities . .4112 , 3035 Hours worked on job 13.9883 12.3509 Grade point average 2.4935 ■ .7315 52 assigned a socio-economic index that was based on prestige, education, desirability, and salary level of each occupation [Reiss, 1961: 263-270] Two problems resulted. First, the out-dated list was based on the 1950 census. Second, many Montana occupations did not relate very well to the list of occupations given. Many other variables showed a strong relationship to each other. Such things as teacher starting salary, per student expenditure, student-teacher ratio, degree level of the teachers, and teacher experience were all related. Other variables, such as community size, the age of the school, and school size, has similar problems. The following describes each of the variables used in the regres­ sion analysis. Each variable is described as to its source and how its value was calculated. Student's View of School This variable is calculated as a simple yes or no response to the question, "Do you like school?" which appears on the questionnaire (see Appendix A). Yes and no were coded as I and 0 respectively. A positive relationship to achievement was anticipated. Per Student Expenditures From the.Office of the Superintendent of Public Instruction, . records of the general fund expenditures for the last three years . (70-71, 71-72, and 72-73) were obtained. The yearly costs were added 53 together and divided by three to give the average yearly expenditure, and this was then divided by the number of students to give the expenditure per student. This variable was predicted to have a positive relationship to achievement. School Size This variable represents the total number of students in each school in all grades. This item was expected to have no effect on . achievement. Father's Educational Level A variable which is represented by the highest grade level the father of the student attained. It was felt that this item would relate positively to achievement. TV, Phonograph, Telephone, Paper This was a weighted item composed by asking each student which items were in the home. The responses were added and divided by four. It was predicted that this variable would be positively correlated to achievement. Books and Magazines Each student was asked the number of books purchased yearly and magazines purchased monthly by the family. These two totals were then added to give a composite. This variable should be positively related to student achievement. 54 Autos in Family The number of automobiles the family owned comprise this variable. A positive relationship was anticipated. Own Room and Car Whether, or not the student had his own room and own car was predicted to have a positive relationship. Each student was asked this question and rated I or 0 then divided by two to give a composite total. ' College Students were asked whether or not they planned to attend college. This did not include business schools, trade schools, or community colleges, but only major four-year schools. The anticipated relation­ ship to achievement on this item was positive. School Activities Each student was asked to specify, in which school activities he participated. The categories he could respond to were: government, athletics, class plays, student organizations, and other. Each answer asked for a yes or no response. These were totaled and divided by five to give the composite score. It.was thought that a positive relationship between this and achievement existed. Hours Worked on the Job If the student had a part-time job outside of school, he was asked how many hours were spent per week working. This was predicted to have I 55 a negative effect on achievement. Grade Point Average This variable represents the grade point average of the student on a four-point scale. Some of the schools gave a simple letter grade and others used a score based on a 100 point scale. Each score was treated separately and was translated to the 4.0 scale at the school. GPA was anticipated to have a positive relationship to achievement. Chapter V RESULTS OF STATISTICAL ANALYSIS The study looked at four dependent variables using multiple regression: English, reading and numerical competency scores and composite score. Each of these variables is a standardized score and this process, as well as a statistical description of the results is provided in previous chapters of this, thesis. Table V-I summarizes the data obtained in the regression analysis for the seniors who were . 2 tested. Each dependent variable is listed with its R value and an F-value. These figures are all senior students only. Table V-2 shows the same information broken down according to the school catagories. 2■The square of the multiple correlation coefficient (R ) shows the pro­ portion of the variance of the dependent variable accounted for by the 2 independent variables. The R varies by school size, suggesting dif­ ferent production functions exist for different size schools. The variables in this study seem to be better predictors for the two intermediate school sizes which range from 118 to 1,040 students than for the largest and smallest schools. Most of the variance can be attributed to factors other than those represented in the.production . function equation. Data indicates that the independent variables used 2 do affect student achievement. The largest R value was obtained by schools with 262-1,040 students. Schools of less than 117 students 2 achieved the smallest R value. When grade point average was included 57 TABLE V-I. . DEPENDENT VARIABLES SUMMARY ALL STUDENTS. Dependent Variable R2 F-Value Reading : with GPA without GPA variable as an independent .38905 . .23331 45.74249 23.87464 English : with GPA without GPA as an independent variable .38719 .21445 45.38680 21.41803 Numerical Competency: with GPA without GPA as an independent variable .36805 .19891 41.83678 19.48079 Total: with GPA .47275 64.40821 without GPA as an independent variable .26271 27.95540 . 58 TABLE V-2. DEPENDENT VARIABLE SUMMARY BY SIZE. With GPA Without GPA Class Size Rz F-Value RZ F-Value 0-117 R eading .33806 4.72407 .24444 3.29402 English .35948 5.19146 .20822 2.67763 Numerical Competency .36763 5.37744 .20866 2.68481 Total .42914 6.95350 .25909 3.56045 188-261 Reading .44339 10.09017 .27256 5.21141 English .44993 10.36087 .30156 6.00541 Numerical Competency .42926 9.52671 .25837 4.84567 Total .53659 14.66676 .32950 6.83535 262-1,040 Reading .48843 29.67770 .30964 15.24068 English .49170 30.06809 .27406 12.83561 Numerical Competency .40500 40.34411 .24638 11.11548 Total .56483 40.34411 .33247 16.93422 1,040 and Reading Up .46196 13.37995 .26237 6.07897 English .45175 12.84053 .23459 5.23812 Numerical Competency .42838 11.67834 .18265 3.81936 Total .52943 17.53228 .25577 5.87365 59 as an Independent variable in the equation, it showed the highest correlation of all the variables to achievement in each instance, the F-value calculated was significant at the .05 level [Snedecor,1967: 561]. Regression Analysis for All Students ' The regression equations shown in this section are those rep­ resenting each school size and representing only senior students. Each equation will first be presented with the pertinent data on each variable; then, each variable will be analyzed individually. The regression equations using the individual test scores as the dependent variable appear in the appendix in Tables I through XXX. The cor­ relation matrix for the variables in the equations appears in Table XXXI. Tables V-3 through V-12 show the equations for the total score achieved by all senior students. The same information is included for each school size with GPA and without GPA as an independent variable. The tables in the appendix contain the same.breakdown for each of the tests administered.' Each table shows the F statistic testing the hypothesis of the regression coefficient being zero. The variables have been marked if they are significant at the 5 percent and 10 per­ cent probability levels based on the F-value. In each equation the Beta value is used to describe the significant variables. The 60 regression coefficient represents numerically the effect that one unit increase in the independent variable has upon the dependent variable. The Beta coefficient shows the results of one standard deviation in­ crease in the independent variable on the dependent variable. For example, if the Beta coefficient was .5 then an increase of one ' standard deviation in the independent variable would cause the depen­ dent variable to increase .5 of its standard deviation. The results of the regression of the total score for all students appear in Table V-3. Grade point average is included in this equation, and has the largest Beta value significant at the .05 level. • From the ata higher GPA correlates positively with higher achievement. The student’s plans for college has the next highest Beta value. This also correlated positively with achievement. This was expected since the student with college intentions probably has a more positive attitude towards learning and school. The third highest Beta value was for per student expenditures. The negative relationship indicates that the more spent per student the lower achievement. This is probably a result of the high per student costs of small schools. The students involvement in school activities, such as student ■ government and athletics, had the next highest Beta value arid also revealed a positive relationship to achievement. This was closely followed by high Beta values related to school attitude and reading material.in the home. Both also had positive relationships to 61 TABLE V-3. TOTAL SCORE EQUATION FOR ALL STUDENTS WITH GPA AS AN INDEPENDENT VARIABLE. Dependent Independent Regression F-Value Total Score GPA 18.40740 343.388* School attitude 2.89286 3.935* Per student expenditure -0.00238 11.034* School size -0.00082 0.455 Father’s education 0.29627 3.419** Items at home 8.89239 . 1.856 Reading material 0.08537 . 5.238* Cars in family 0.83606 2.131 . Own room and car -2.82010 1.910 College intention 8.25217 30.921* School activities 5.85408 5.265* Hours worked -0.04515 0.902 Constant 92.21983 R2 = .47275 E-value - 64.40821* *Signifleant at the 5 percent level. **Signifleant at the 10 percent level. 62 achievement. This was closely followed by high Beta values related to school attitude and reading material in the home. Both also had positive relationships to achievement. All of the above variables were significant at the .05 level. Two other variables were significant at the .05 level. Two other variables were significant at the .10 level. They were father's educational level and the number of cars in the family; both of which showed positive relationships to achievement. They were school size, items in the home, whether or not the student had their own room and car, and hours working on an outside job. Table V-4 is the total score equation for all students without grade point average as an independent variable. The highest Beta value was for college intentions which had a positive relationship to achievement. The next highest Beta value was related to involvement in achool activities and this also revealed a positive relationship.. The third highest Beta value was obtained by the school attitude variable, followed in order by: per student expenditures, student's own room and car, father's educational level, reading material in the home, and hours worked on a job outside of school. Three of the variables had a negative relationship to achievement. These were per student expenditures, own room and car, and hours spent on an outside job. All of the variables mentioned here were significant at the .05 level.' Three variables in the equation were not significant; these were school size, items in the home, and cars in the family. 63 TABLE V-A. TOTAL SCORE EQUATION FOR ALL STUDENTS WITHOUT GPA AS AN INDEPENDENT VARIABLE. Dependent Variable Independent Variable Regression Coefficient F-Value Total Score School attitude 6.89409 16.358* Per student expenditure -0.00285 11.309* School size 0.00049 0.115 . Father’s education 0.41821 4.885* . Items at home 10.58202 1.882 Reading material 0.08751 3.941* Cars in family 0.83606 2.131 . Own room and car -5.18910 4.649* College intentions 15.80306 87.807* .School activities 14.51858 23.989* Hours worked -0.14737 6.975* Constant 128.84119 R2 = .26271 F--value - 27.9554* *S.ignifleant at the 5 petcent level. **Slgniflcant at the 10 percent level. 64 Regression By School Size Tables V-5 and V-6 show the regression equations for schools with . 0-117 students. Table V-5 represents the equation which includes grade point average as an independent variable. Only two variables were significant at the .05 level, GPA, which has the higher Beta value, and the number of cars in the family. Two other variables, were significant, at the .10 level. The highest Beta value of these was for own rpom and own car followed by the student|s involvement in school activities. Grade point average, cars in the family, and school activities all demonstrated a positive relationship to achievement. - If the student has his own room and car it showed a negative relation­ ship. All other variables - school attitude, per student expenditure, school size, father's educational level, items at home, home reading material, college intentions, and hours worked - showed no significant ' relationship to achievement. When GPA was dropped from the equation the student's college intentions, participation in school activities, and the number of cars in the family had the highest Beta values respectively. College intentions and involvement in school activities were both significant at the .05 level. Only cars in the family was significant at the .10 level. School activities are important in a small school probably because of the student population; therefore, the number of activities one is involved in should reflect positively 65 TABLE V-5. TOTAL SCORE EQUATION FOR SCHOOL SIZE 0-117 STUDENTS. Dependent Variable Independent ' Variable Regression Coefficient F-Value Total Score GPA 16.95502 33.064*, School attitude 1.04790 0.062 Per student expenditures 0.00165 . 0.220 School size 0.13213 0.484 Father’s education . -0.25710 0.234 Items at home -5.78317 0;100 Reading material 0.01395 0.026 Cars in family 2.73050 4.972* Own room and car -11.53994 3.813** College intention 4.45649 1.089 School activities 11.99985 2.814** Hours worked -0.02264 0.033 Constant 92.08910 R^ = .42914 F-value - 6.95350* *Signifleant at the 5 percent level. **S±gniflcant at the 10 percent level 66 TABLE V-6. TOTAL SCORE EQUATION FOR SCHOOL SIZE 0-117 WITHOUT GPA. Dependent Variable Independent Variable Regression Coefficient F-Value Total Score School attitude 5.05201 1.154 Per student expenditures -0.00039 0.010 School size 0.15929 0.547 Father's education -0.31326 0.247 Items at home -7.87460 0.144 Reading material 0.02000 0.041 . Cars in family 2.47930 3.191** Own room and car -6.38240 0.928 College intention 13.82656 9.538* School activities 17.04811 4.483* Hours worked -0.12536 0.794 Constant 129.26440 R = .25909 F-value - 3.56045* ^Significant at the 5 percent level. **Signifleant at the 10 percent level. 67 on school, attitude. Schools with 118-261 students are represented in Tables V-7 and V-8. Table V-7 includes GPA as an independent variable.. Three : variables showed significance and all were significant at the .05 level. GPA was the only one of these which showed a positive relation­ ship. It also had the highest Beta value. Per student expenditures had the.next highest relationship to achievement. When GPA. was dropped as an independent variable (Table V-8) four variables showed significance at the .05 level and one at the .10 level. The highest Beta value was school size, it was followed by per student expendi­ tures. Both of these variables correlated negatively with achievement. College intentions had the third highest Beta value and school attitude the fourth highest. Both of these showed a positive relationship to achievement and both were significant at the .05 level. The single variable significant at the .10 level was.the students involvement in school activities. It was also positively related to school achieve­ ment . Tables V-9 and V-10 show the equations for schools with 262-1,040 students. Table V-9 has grade point average as an independent Variable and in Table V-10 it does not appear in the equation. In the equation ■ shown with GPA four variables are significant at the .05 level. The highest Beta value is for GPA. This is followed by college intentions, and reading material in the home. All three of these variables showed 68 TABLE V-7. TOTAL. SCORE EQUATION FOR SCHOOL SIZE 118-261 STUDENTS. Dependent Variable Independent Variable Regression Coefficient F-Value Total Score GPA 17.486.55 67.924* School attitude 4.32618 1.814 Per student expenditures -0.00728 28.443* School size -0.18469 17.822* Father’s education -0.60583 2.416 Items at home 2.33934 0.038 Reading material -0.01980 0.083 Cars in family■ 0.53841 ‘ 0.196 Owns room and car 3.56811 0.673 College intention 4.50904 1.592 School activities -1.93290 0.120 Hours worked 0.07519 0.458 Constant 147.93963 O . R = .53659 F-value - 14.66676* *Signifleant at the 5 percent level. 69 TABLE V-8. TOTAL SCORE, EQUATION FOR SCHOOL SIZE 118-261 WITHOUT GPA. . Dependent Variable Independent Variable Regression Coefficient F-Value Total Score School attitude 9.02812 5.676* Per student expenditure -0.00600 13.637* School size -0.19363 13.636* Father’s education 0.63353 1.838 Items at home 6.71801 0.216 Reading material 0.03826 0.217 Cars in family 0.47610 ■ 0.107 . Own room and car 1.47132 0.080 College intention 15.75625 15.873* School activities 11.31572 3.129** Hours worked -0.07364 0.314 Constant 171.92228 ST A EMTNOF PRIvalue - 6.83535* . ^Significant at the 5 percent level. **Significant at the 10 percent level. 70 TABLE V-9. TOTAL SCORE EQUATION FOR SCHOOL SIZE 262-1,040 STUDENTS. Dependent Variable Total Score Independent Regression Variable . Coefficient • ■ F-Value GPA School attitude Per student expenditure School size Father's education Items at home Reading material. Cars in family Own room and car College intention . School activities Hours worked Constant R2 - .56483 21.97017 199.158* . 1.59499 ■ 0.549 0.00410 1.882 0.00755 2.205 0.26865 1.551 17.67634 3.646** 0.24809 16.967* 0.46048 0.477 -4.93454 2.890** 10.64080 27.062* ' 4.12334 ■ 1,189 -0.16747 5.637* 52.93562 F-value - 40.34411* *Signifleant at the 5 percent level. **Signifleant at the 10 percent level. 71 TABLE V-IO. TOTAL GPA. SCORE EQUATION FOR SCHOOL SIZE 262-1,040 WITHOUT Dependent Variable Independent Variable Regression Coefficient F-Value School attitude 8.35189 10.353* Per student expenditure 0.00244 0.437 School size 0.01033 2.705** Father's education 0.50986 3.674** Items at home 20.39174 3.173** Reading material 0.23532 9.980* Cars in family 0.08604 0.011 Own car and room -10.53170 8.768* College intention 15.05602 36.263* School activities 16.75660 13.595* Hours worked -0.24154 7.707* Constant 102.02085 . R2 = .33247 . .E-value - 16.93422* *Significant at the 5 percent level. **Signifleant at the 10 percent level. 72 a positive relationship to achievement. The number of hours worked had a negative relationship to achievement and the fourth highest Beta value. The next two highest Beta values were for variables that were significant at the .10 level. Items in the home had the highest Beta value and was positive in its relationship to achievement. If the student had his own room and car, the correlation to achievement was negative. All other variables showed no significant relationship. Table V-IO had only two variables not significant at the .05 or .10 level. School attitude, reading material in the home, if the student had their own room and car, college intentions, involvement in school activities,, and hours worked were all significant at the .05 level. Two variables, school size and father's educational level, were, significant at the .10 level. The highest Beta value was for the college intentions of the studeiit. The variables seemed to show the most reliable relationship to achievement in schools of this size. The equations for schools with an enrollment over 1,040 students enrollment is contained in Tables V-Il and V-12. The equation.with grade point average as a variable (Table V-ll) had only four .signifi­ cant, variables all at the .05 level. GPA had the highest Beta value and a positive relationship to achievement. The second highest Beta was for the negatively related per student expenditures. Items in the home and college intentions, both with a positive relationship to achievement, had the next highest Beta values. No other variables in 73 TABLE V-Il. TOTAL SCORE EQUATION.FOR SCHOOL SIZE OVER 1,040. Dependent Variable Independent Variable Regression Coefficient F-Value Total Score GPA .21.12409 108.747* School attitude -2.78054 0.790 Per student expenditure , -0.00577 9.864* School size 0.00932 2.199 Father's education 0.40655 1.493 Items at home 74.29762 9.381* Reading material -0.05678 0.364 Cars in family -1.09659 0.909 Own room and car 4.53840 1.018 College intention 7,87742 6.212* School activities 0.03460 0.000 Hours worked 0.01500 0.023 Constant 18.77307 R^ = .52943 F-value - 17.53228* ^Significant at the 5 percent level. ^Significant at the ID percent level. 74 TABLE V-12. TOTAL SCORE EQUATION FOR SCHOOL. WITHOUT GPA. SIZE 1,040 AND UP . Dependent Independent Regression Variable Variable Coefficient F-Value Total Score School attitude -1.03101 0.069 Per student expenditure -0.00508 4.877* School size 0.00233 0.089 Father's education 0.80128 3.736** Items at home 60.56828 3.975* Reading material -0.11956 1.030 . • Cars in family -0.09141 0.004 Own room and car -0.60048 0.011 College intention 16.87547 19.580* School activities 9.24075 1.371 Hours worked -0.08679 0.493 - Constant . 84.46571 R^ = .25577 F-value =■ 5.87365* . *Signifleant at the 5 percent level. ^^Significant at the 10 percent level. 75 this equation showed a significant relationship to achievement. When GPA is not 'a variable, the equation (Table V-12) shows that college intentions had the highest Beta value and is positively related to achievement. The second highest Beta was for per student expenditures, a negatively related variable. The Beta value for items in the home ranked third, and this was a positive relationship to achievement. These three variables were significant at the .05 level. One variable, father's education, was significant at the .10 level and it showed a positive relationship to achievement/ No other variables revealed any significant relationship to achievement. Grade point average is significant in every equation and it correlates positively with achievement. The GPA may be'dependent on many different items such as student motivation, student's intelligence, and parental attitudes. Table V~13 shows the' correlation coefficients of grade point average and the English ,■ numerical competency, and total scores. ' Although the relationships are high it is felt that GPA is important [Draper,1966; 147-150]. Using the data collected in this study, regression analysis was run with GPA as the dependent variable and the same independent variables as before. The equation is shown in Table V-14. Eour of the variables with the highest Beta values from high to lew were college intentions, school attitude, involvement ■ 76 TABLE V-13. GPA CORRELATION COEFFICIENTS WITH THE DEPENDENT VARIABLES English Numerical Competency Reading . Total GPA ■ .65 .56 .61 .68 TABLE V-14. THE EQUATION WITH GPA AS THE DEPENDENT VARIABLE. Dependent Independent Regression Variable Variable Coefficient F-Value School attitude 0.21737 • 19.342* College intentions 0.41021 70.369* School activities 0.47071 29.991* Hours worked ■ -0.00555 11.779* Own room and car -0.12870 3.402** School size 0.00007 2.912** Per student expenditure -0.00003 1.064 Father's education 0.00662 1.458 Home items ■ 0.09179 0.168 Reading material 0.00012 • 0.008 Cars in family 9 . 0.00128 0.006 R2'= .23831 F-value - 24.5457 ^Significant at the 5 percent level. **Significant at the 10 percent level 77 in school activities, and hours worked. Two variables were significant at the .10 level. These were own room and car, which had a negative relationship, and school size which was positively related. No other variables showed significance. The variables only explain 23 percent ■ 2 of. the variation in R . The student's .attitude toward school shows a positive relationship for all students. In cases where it is related to achievement by school size, its relationship varies in that it is not always a ,significant variable. This seems to show a different production function for different size schools. Per student expenditure again is not always a significant variable and when significant it shows a. negative relationship to achievement. Its relationship in those cases is small as its regression coefficient is never.larger than a negative .01. School size is not significant in most cases' either. It.shows significance only in the schools of size 118-1,040 student population ■ and then,;only if GPA is not an independent variable. Its relationship in these cases is positive. Father's education shows significance only when viewed for all students and in larger schools. This may be a result of lower education levels in rural areas. Its relationship is positive. Father's educational level is used as a proxy for the socio-economic status of the family. It was felt to be the best predictor based on 78 other studies and upon its relationship in these equations when com­ pared to a socio-economic index on the mother's education level. Items in the home shows significance only in some of the size classifications. It is a positive relationship to achievement. Reading material has a positive relationship in all cases where it shows significance. It does not seem related in the.smaller more rural schools. Cars in the family was used as an indicator of economic status. It is significantly related in only three cases and two of those are at the 10 percent level of significance. Its relationship, when significant, is positive. I Whether the student has his own room and car or not shows signifi­ cance in some cases. The relationship to achievement is negative in these cases. The college intentions of the students is significant in most. cases. Only in the two smallest school size classifications does it not show significance and then only if GPA is not an independent variable. Its relationship in each case is positive. .The student's involvement in school activities is important in most cases. It .shows a positive relationship to achievement where it is significant. In the larger school classes it is not as consistently significant. 79 •The amount of the time a student spends working at an outside job is significant in three cases when looking at the total score equations. This relationship in all instances is negative. The equations do show marked differences of significant variables by school, size. The importance of the common variables in these cases shows some variation as well. In all equations the only items that show up consistently are GPA» home items, and per student expenditures Not in all cases is per student expenditures a positive relationship for the individual tests. This also varies by school size. Chnpteir VI SUMMARY AND CONCLUSIONS This study has attempted to look at Montana’s educational system and its impact on the student, both rural and urban. The' researcher has tried to make a determination of the educational quality in Montana schools. This chapter will first summarize the study and secondly present those conclusions that can be made from the study. . . Summary . The study had two major objectives: ■ I) to analyze whether or not school size is important in determining student achievement; and 2) to analyze factors that affect student achievement and to determine whether or not these factors vary in different size schools. The major steps in accomplishing the objectives were as follows: First, samples of schools and students were taken with the aid of the Mathematics Department at Montana State University. Time and money limited the sample to about 10 percent of the sophomores (grade 10) and 10 percent of the seniors (grade 12) in the state. Out of approximately 50,000 high school students in Montana, 2,186 were tested, including 1,311 sophomores and 7 percent of the seniors. The second step was to design the necessary collection instruments and select an 81 achievement test. The Stanford Achievement Test (SAT) was chosen on a basis of price availability and a recommendation by Buroa [1972]. Third,, each school was visited and the students were tested by the data collectors hired for the project. In the fourth step the data .. were verified, checked and coded for input into the computer. The analysis utilized packages by Lund [1973] and a system called SPSS [Nie,1970]. Analysis of variance and regression analysis were the two major statistical techniques used in the analysis. Finally,' the data collected were analyzed in light of the project objectives. Conclusions The first of the objectives was to analyze whether or hot.school size is important in determining student achievement. The results of the statistical analysis in. Chapter IV of this study address this objective. The mean score of each test was calculated for each school. This score was standardized so that a composite total score might also be calculated. This procedure was the same for both sophomores and seniors. The means for the seniors were ranked from highest to lowest and an analysis of variance procedure was used to test mean' differences. The range of the means was about 13 points for .English, 15 points for reading, 13 points for numerical competency, and 20 points for the composite total score. The analysis of variance procedure 82 showed a significant difference at the .05 level of significance. The Scheffe test was used to compare the individual means. A comparison of the highest to lowest indicated no significant difference. This left the conclusion that a significant difference does exist between the highest and lowest mean hut no other comparison could be made with the methods used in the study. A comparison of the means was also made by school size. In this case the analysis of variance procedure showed a significant difference only in the case of numerical competency. Again the Scheffe test was used and no significant difference was found between the means: The ranges of means in all test scores by school size was less than three points. On the basis of the statistical analysis applied, no significant difference could be determined. The same procedures were used on the mean scores of the sophomores. The analysis of variance showed a difference in mean scores at the .05 level of significance but the Scheffe test showed ho significant difference between the highest and lowest test scores. When the mean scores were ranked from highest to lowest the ranges of the means were 11 points for the English test and the reading test, 10 points on the numerical competency test, and 18 points for the total.score. A comparison of the means for each size classification showed significant differences in the means among total score, English, and reading. In each of these cases the 83 highest mean was for large schools and the lowest mean was for small schools.. The Scheffe test showed no significant difference between the highest and lowest means. Based on these results it seems reasonable to conclude.that at the senior level in high school no significant differences exist in the mean test scores based on size of the school. This is not true at the sophomore level of high school. The sophomores of the larger high schools do seem to score better on achievement tests than do those from the smaller rural schools. When the means of the individual schools are compared > the seniors' achievement is fairly evenly interspersed throughout rural and urban schools. The ranking of the schools by sophomore mean score seems to place more urban schools than rural schools at the top of the scale. A comparison of the test scores to SAT norms reveals interesting results. The means of the seniors and sophomores in every test.fell in the fifth stanine ranking which is average for all students taking the test. The percentages of students below the fifth stanine by school size showed only one group with more than 50 percent of the students at the fourth stanine or below. In schools with less than 118 students in size, 54 percent of the seniors tested fell at the fourth stanine or below. In each test \ administered and for each school size the seniors placed a larger percentage of students below the fifth stanine. than did the 84 sophomores. In general the sophomores did much better than seniors. This might be a result of the sophomores being closer to actual course work in the subject areas tested. From the results, however, it seems that Montana students on the whole do fairly well when compared to the national performance statistics on this particular achievement test. The sophomores show up very well on such a comparison and the seniors place about average. The sophomores in the largest schools place better than do the sophomores in the other three size classifications. The seniors do not seem to vary much from one size class to another. The second objective was to analyze factors that affect student achievement and to determine whether or not these factors vary-in different size schools < Regression analysis was' used to analyze .the, data for this objective. Twelve variables were chosen to represent the school, the student, and socio-economic status. The independent variables were regressed against each of the dependent variables for each of the four school size categories and for all students sampled. The regressions for all students as a group show GFA, school attitude, per student, expenditure, reading material in the home, ' college intentions, and extracurricular school activities to be the most important variables in determining student achievement.- School size was not a significant variable. When GPA was dropped as a 85 variable three other variables, father's education, own. room and car, and number of hours the student worked at a job became significant. In both cases per student expenditures had a negative effect on achievement. It appears that student measures such as GPA, school attitude, and socio-economic measures such as reading material and college intentions have the strongest effect on student achievement. When looking at the equations as they apply to the different size schools it is obvious that the factors differ in their relationship to student achievement. In the smallest rural schools only two variables are strongly significant, GPA and cars in the family. When GPA is omitted as a variable, college intentions and extracurricular school activities become significant. The largest size class equation shows four significant variables, GPA, per student expenditures, items in the home, and college intentions. Omitting GPA changes the influence of the father's educational, level. Per student expenditure has a negative effect on achievement in both equations. The other two size classes show much stronger ■ relationships to the variables. Only in schools of size 118-261 students is school size a significant.variable. In schools, of 262-1,040 students neither school size nor per student expenditures are significant. Hours worked at an outside job show a negative relationship to achievement in this size school. 86 The variables do seem to vary in their relationship according to school size. The rural schools seem to have different variables ■ influencing student achievement than the large urban schools. Socio-economic backgrounds of the students seem to have the most influence on achievement. There does seem to be a relationship between school inputs and socio-economic status. Both affect achievement. This study does not provide an answer to the problems of school productivity. It does suggest that there is something more to increasing output quality than increasing funding. It does support some ideas that different schools of the same size may have to spend different amounts to maintain achievement at a consistent level. There are variables beyond the control of the school system that effect student achievement which must be taken into consideration in estimating a school production function. The findings highlight the need for continuous efforts to specify the relevant outputs of the educational system as well as the optimum input levels. Such information is crucial to good decision making with respect to human resource development. The findings of this study suggest additional research which lies beyond the capabilities of the present data. The affects of school inputs on social backgrounds and the production functions of school administrators with emphasis on budget constraints and legal 87 responsibilities warrant further investigation. A definitive measure of achievement might be gained by testing the student on entrance and exit from the school to obtain the differences in achievement. Other measures of achievement .should be considered; student self-concept and physical development are examples. A more definitive view of the effects of schools in the education process is needed. A number of production functions should be considered simultaneously rather than a single equation. In conclusion, the rural student seems to do about as well as the urban student, yet the variables influencing achievement differ for the rural student. The results may indicate possible policy implications. The data seems to conclude that most high school students are. disadvantaged by attending the last two years of high school. In terms of our. measures of achievement the sophomores show generally higher levels of achievement than do the seniors. Based on this it is. possible that students should be channeled into university or other vocational training at the end of the tenth grade. This also casts doubt on aid to schools from the State or Federal.governments for grades eleven and twelve. The lack of differences in achievement between large and small schools also opens areas for school consolidation. If no significant, difference in achievement is evident then economies of scale might be accomplished through school consolidation. The acceptance of the 88 idea that different combinations of inputs are required for different schools may have implications in the area of school finance. If a- school, to achieve, a desired level achievement must use input factors that.are more capital intensive than a school of equal size then equal educational opportunity is not achieved . through equal expenditures. APPENDIX 90 TABLE I Reading Equation For All Students Dependent Independent Regression Variable Variable Coefficient . F-Value Total Score GPA '■ 6.27895 219.727 **' School Attitude 1.31948 4.502 ** Per Student Expend. - .00096 9.787 ** School Size .00016 .093 Father's Education .13445 3.872 ** .Items at Home 1.36068 .239 Reading Material .02373 2.226 Cars in Family .36131 •3.058 Own Room and Car - .47539 . ! .299 College Intention 3.31108 27.376 ** School Activities 1.38259 1.615 ' Hours Worked - .06959 11.785 Constant 31.46214 R2 = .38905 F-value = 45.74249 ** * = Significant at the 10% level. ** - Significant at the 5% level. 91 TABLE II Numerical Competency Equation For All Students Dependent Independent Regression Variable Variable Coefficient F-Value Numerical GPA 6.03722 • 230.713 ** Competency School Attitude 1.08709 3.401 * Per'Student Expend. - .00080 7.840 A A School Size .00078 2.539 Father's Education .06293 .964 Items at Home 4.27690 2.682 * . Reading Material. .02315 2.406 Cars in Family .52933 7.454 A A Own Room and Car - .34111 .175 College Intention 2.68008 20.371 A A School Activities 1.75554 . 2.957 A Hours Worked .04442 5.454 A A Constant 29.22282 R2 = .36805 F-value = 41.83678 ** * = Significant at the 10% level. ** = Significant at the 5% level. 92 TABLE III English Equation For All Students Dependent Variable Independent Variable Regression Coefficient ■ Fr-Value English GPA 6.09124 242.982 ** School Attitude .49729 .751 Per Student Expend. - .00062 4.862 ** School Size - .00020 .182 Father's Education ..09889 2.461 Items at Home 3.25481 ■ 1.607 . Reading Material .03849 CEYYT -n Cars in Family - .07808 . .168 " Own Room and Car . -2.00360 6.232 *A . College Intention 2.26101 .15.000 ** School Activities 2.71594 pEMTT -- Hours Worked - .01998 1.142 Constant 31.53488 R2 = ,38719 F-value = 45.38680 *A * = Significant at the 10% level. ** = Significant at the 5% level.. 93 TABLE IV Reading Score For All Students Without GPA Dependent Variable Independent Variable Regression Coefficient F-Value Reading School Attitude 2.68434. 15.197 **. Per Student Expend. - .00112 10.633 ** School Size .00060 1.087 Father’s Education .17604 ■ .5.304 ** Items at Home 1.93703 .387 Reading Material ' . .02446 1.887 . Cars in Family .36933 2.549 Own Room and Car -1.28348. ■ . 1.743 College Intention 5.88676 74.665 ** School Activities 4.33814 13.125 ** Hours Worked - .10446 21.473 ** Constant 43.95415 R2 = .23331 F-value.= 23.87464 ** * = Significant at the 10% level. ** '= Significant at the 5% level. 94 .TABLE V Dependent Independent Regression Variable Variable ■___ Coefficient F-Value Numerical Competency Equation For All Students Without GPA Numerical School Attitude 2.38840 13.527 ** Competency Per Student Expend. - .00096 . 8.784 ** School Size - .00035 .403 Father's Education .10293 2.039 Items at Home 4.83106 2.703 * Reading Material- .02385 2.017 Cars in Family .53704 6.059** Own Room and Car -1.11809 1.487 . College Intention 5.15661 64.-416, ** School Activities 4.59731 16.573 ** Hours Worked .01089 .263 Constant 41.23390 R2 = .19891 F-value = 19.48079 ** * = Significant at the 10% level. ** = Significant at the 5% level. 95 Dependent Independent Regression Variable_____ Variable________ Coefficient F-Value TABLE VI English Equation For All Students Without GPA School Attitude 1.82135 8.048 ** Per Student Expend. - .00078 5.922 ** School Size .00023 .179 Father’s Education .13924 3.818 * Items at Home 3.81393 1.724 Reading Material .03920 5.575 ** Cars in Family - .07031 .106 Own Room and Car -2.78753 9.458 ** College Intention 4.75969 56.148 ** School Activities 5.58313 25.007 ** Hours Worked - .05381 6.555 ** Constant 43.65343 R2 = .21445 F-value = 21.41803 ** * = Significant at the 10% level. ** = Significant at the 5% level. 96 TABLE VII Reading Equations For School Szie 0-117 Students Dependent Variable Independent Variable Regression Coefficient F-Value Reading GPA 5.07472 15.699 ** School Attitude - .69939 .147 Per Student Expend. .00079 .269 School Size .07846 .904 Father's Education - .18537 .645 Items at Home -3.51020 .195 ■ Reading Material .01059 .078 Cars in Family .91400 2.953 * Own Room and Car -3.48222 1.840 College Intention 2.99.685 ■ 2.610 School Activities 5.50394, 3.138 * Hours Worked - .02273 .174 Constant 28.82238 R2 = .33806 F-value = 4,.72407 ** * = Significant at the 10% level. ** = Significant at the 5% level. 97 TABLE VIII Numerical Competency Equation For School Size 0-117 Students Dependent Independent Regression Variable Variable Coefficient F-Value Numerical GPA 6.03663 • . 27.903 ** . Competency School Attitude 1.96311 1.451 Per Student Expend. .00109 .647 School Size .02715 .136 Father's Education .03795 .034 Items at Home .33672 .002 Reading Material - .01471 .189 Cars in Family •1.37295 8.369 ** Own Room and Car -2.29986 1.008 College Intention .31120 • .035 School Activities 3.68368 1.765 Hours Worked .02759 .322 , Constant 23.56233 • R2 = .36763 . F-value = 5.37744 ** * = Significant at the 10% level. ** = Significant at the 5% level. 98 TABLE IX English Equation For School Size 0-117 Students Dependent Independent Regression Variable Variable Coefficient F-Value English GPA 5.84367 26.213 ** School Attitude - .21581 .018 Per Student Expend. - .00024 .031 School Size .02651 .130 Father's Education - .10968 .284 Items at Home -2.60970 .136 Reading Material .01807 .286 Cars in Family .44355 .876 Own Room and Gar -5.75785 6.336 ** College Intention I.14844 .483 School Activities . 2.81224 1.031 Hours Worked - .02750 .321 Constant 39.70439 = .35948 F-value = 5.19146 ** * - Significant at the 10% level. ** = Significant at the 5% level. 99 Dependent Independent Regression Variable . Variable________._____ Coefficient F-Value TABLE X Reading Equation For School Size 118-261 Students GPA 6.08486 46.652 ** School Attitude 2.31358 2.943 * Per Student Expend. - .00258 20.263 ** School Size - .06277 11.675 ** Father's Education .20299 1.539 Items at Home 1.88895 .139 Reading Material - .02260 .610 Cars in Family .72472 2.019 Own Room and Car 1.19526 .429 College Intention .72644 .234 School Activities '-2.01628 .743 Hours Worked - .03825 .673 Constant 48.42904 R2 = .44339 F-value = 10.09017 ** * = Significant'at■the 10% level. ** = Significant at the 5% level. 100 TABLE XI Numerical Competency For School Size 118-261 Students Dependent Independent Regression Variable Variable Coefficient F-Value Numerical GPA 5.83205 45.511 ** Competency School Attitude .95142 .529 Per Student Expend. - .00207 13.902 ** School Size - .03582 4.038 ** Father’s Education .14944 .886 Items at Home -2.78007 .319 Reading Material - .01592 .322 Cars in Family .31401 .403 Ovm Room and Car 1.73788 .962 College Intention 2.71106 3.467 * School Activities - .71768 .100 Hours Worked .07344 2.634 Constant 45.66785 R2 = .42926 F-value = 9.52671 * r» Significant .at the 10% level. ' ** - Significant at the 5% level. . 101 Dependent Variable TABLE XII English Equation For School Independent Variable. Size 118-261 Regression Coefficient Students F-Value English GPA. 5.56964 41.000 ** School Attitude 1.06118 .650 Per Student Expend. - .00263 . 22.026 ** School Size - .08611 23.047 ** Father's Education .25340 ' 2.515 Items at Home 3.23046 .426 Reading Material .01871 .439 Cars in Family - .50032 1.009 Own Room and Car . .63497 .127 College Intention 1.07154 . .535 School Activities .80105 .123 Hours Worked .04001 . .772 Constant 53.84273 R2 = .44993 F-value = 10.36087 ** * = Significant at.the 10% level, ** ='Significant at the 5% level. 102 TABLE XIII J 1 Dependent Independent Regression Variable Variable ____ • Coefficient F-Value Reading Equation For School Size 262-1040 Students GPA 7.66077 130.364 ** School Attitude 1.09628 1.396 Per School Expend. .00137 ‘ 1.133 School Size .00600 7.499 ** Father’s Education ..07915 .725 Items at Home 5.00911 . 1.576 Reading Material .10519 rCEeTr -n Cars in Family .14576 .257 Own Room and Car -1.14422 .837 College Intention 4.09884 21.618 *A School Activities .40427 .062 Hours Worked - .12572 17.104 ** Constant 15.55957 R2 = .48843 F-value =.29.67770 ** * = Significant at the 10% level. ** = Significant at the 5% level. 103 TABLE XIV Numerical Competency Equation for School Size 262-1040 Students Dependent Variable Independent Variable Regression Coefficient. F-Value Numerical GPA 6.61242 99.439 ** Competency School Attitude .23532 .066 Per Student Expend. .00058 .206 School Size .00233 1.160 Father's Education .11384 1.535 Items at Home 8.62496 4.785 ** Reading Material .08188 10.186 Cars in Family .34934 1.513. Own Room and Car -1.85082 2.241 College Intention 3.22754 13.723 ** School Activities 2.51048 2.429 Hours Worked .02409 .643 Constant 18.60088 ■ R2 = .40500 F-yalue = 21.15768 ** * = Significant at the 10%.level. ** -Significant at the 5% level. 104 TABLE XV English Equation For School Size 262-1040 Students Dependent Independent Regression Variable Variable Coefficient F-Value English GPA 7.69698 159.710 ** School Attitude .26339 .098 Per Student Expend. .00215 3.386 * School Size - .00078 .155 Father’s Education .07566. .804 Items at Home 4.04227 . 1.246 ' Reading Material .06102 6.707 ** Cars in Family .03461 .018 Own Room and Car -1.93949 2.917 * College Intention 3.31442 17.155 ** School Activities 1.20858 .667 Hours Worked - .06583 5.691 ** Constant 18.77517 R2 = .49170 F-value = 30.06809 ** * “ Significant at the 10% level. ** = Significant at the 5% level. 105 TABLE XVI Reading Equation For School Size Over 1040 Dependent Variable Independent Variable Regression Coefficient F-Value Reading GPA 7.19306 69.372 ** School Attitude - .98970 .551 Per Student Expend. - .00183 5.454.** School Size. .00206 .593 . Father's Education .39162 7.624 ** Items at Home 14.40200 1.939 Reading Material - .08142 4.117 ** Cars in Family - .20852 .181 Own Room and Car 2.34830 1.499 College Intention 3.45569 6.577 ** School Activities - .59717 .049 Hours Worked - .04973 . 1.387 Constant 15.00769 R2 = .46196 F-Value = 13.37995 ** * - Significant at the 10% level. ** = Significant at the 5% level. 106 . TABLE XVII. Numerical Competency For School Size Over 1040 Dependent Independent Regression Variable Variable Coefficient .■F-Value' Numerical GPA 7.18061 80.386 ** Competency School Attitude - .06345 .003 Per Student Expend. - .00221 9.247 ** School Size .00423 2.901 * Father's Education - .11043 ■ . 705 Items at Home 21.91025 5.219 ** Reading Material .00997 .072 Cars in Family .16358 .129 Own Room and Car 2.52569 2.017 College Intention 2.80872 5.052 ** School Activities - .11500 .002 Hours Worked .04391 1.258 Constant 6.79256 . R2 = .42838 F-value = 11.67834 ** * = Significant at the 10% level. ** = Significant at the 5% level. I 107 Dependent Variable TABLE XVIII English Equation For School Size Over 1040 Independent Variable Regression Coefficient F-Value English GPA 6.75043 E peEFpT -n School Attitude -1.72739 2.035 Per Student Expend. - .00173 5.929 ** School Size .00302 1.544 Father's Education .12535 .947 ' Items at Home 37.98537 16.355 ** Reading Material .01467 .162 Cars in Family -1.05166 5.576 *A Own Room and Car - .33559 .037 College Intention 1.61302 1.737 School Activities .74678 .092 Hours Worked. .02082 .295 . Constant -3.02717 R2 = .45175 F-value = 12.84053 *A * = Significant at the 10% level. ** = Significant at the 5% level. i 108 TABLE XIX Reading Equation for School Size 0-117 Without GPA Dependent Variable Independent Variable Regression Coefficient F-Value Reading School Attitude .49906 .068 Per Student Expend. .00018 .012 School Size .08659 .974 Father's Education - .20218 .678 Items at Home -4.13617 .239 Reading Material .01240 .095 ■ Cars in Family .83882 2.201 Own Room and Car -1.93855 .516 College Intention 5.80135 10.119 *A School Activities 7.01490 ' 4.574 f f Hours Worked - .05347 .870 . Constant 39.94913 R2 = .24444 E-value = 3.29402 ** * = Significant at the 10% level. -n = Significant at the 5% level 109 TABLE XX Numerical Dependent Variable Competency Equation for Independent Variable School Size 0-117 Without GPA Regression Coefficient F-Value Numerical School Attitude 3.38872 3.583 * Competency Per Student Expend. .00037 .059 School Size .03682 .202 Father's Education .01796 .006 Items at Home - .40790 .003 Reading Material - .01255 .111 Cars in Family 1.28351 5.905 ** Own Room and Car - ,46359 .034 College Intention 3.64730 4,582 ** School Activities 5.48105 3.199 * Hours Worked - .00898 .028 Constant 36.79815 R2 = .20866 F-value = 2.68481 ** * = Significant at the 10% level. ** = Significant at the 5% level. H O • TABLE XXI English Equation For School Size 0-117 Without GPA Dependent Variable Independent Variable Regression Coefficient F-Value English School Attitude 1.16423 .429 Per Student Expent. - .00094 .396 School Size .03587 .194 Father's Education - .12903 .321 Items at Home . -3.33052 .180 Reading Material .02015 .290 Cars in Family .35698 .464 Own Room and Car -3.98027 2.530 College Intention 4.37790 6.700 ** School Activities 4.55216 2.240 Hours Worked - .06291 1.400 Constant 52.51712 R2 = .20822 F-value = 2.67763 A* * = Significant at the 10% level. A* = Significant at the 5% level. Ill TABLE XXII ' Reading Equation For School Size 118-261 Without GPA Dependent Variable Independent Variable Regression Coefficient F-Value Reading School Attitude 3.94973 6.823 ** Per Student Expend. - .00214 10.838 ** School Size - .06588 9.911 ** Father's Education .21263 1.300 ■ Items at Home 3.41261 .350 Reading Material - .00239 .005 Cars in Family .70304 1.463 ' Own Room and Car .46563 .050 College Intention 4.64017 8.625 ** , School Activities ,2.59389 1.032 Hours Worked - .09004 2.950 * Constant 56.77437 R2 = .27256 F-value = 5.21141 ** * = Significant at the 10% level. ** ^ Significant at the 5% level. 112 TABLE XXIII Numerical Competency Equation For School Size 118-261 Without GPA Dependent Independent Regression Variable Variable Coefficient F-Value Numerical School Attitude 2.51959 2.965 * Competency - ■ Per Student Expend. - .00165 CEYNM -n School Size . - .03880 3.673 * Father's Education . .15868 .773 Items at Home -1.31971 .056 Reading Material .00345 .012 . Cars in Family .29323 . .272 Own Room and Car 1.3857 .267 . College Intention 6.46219 17.867 ** School Activities 3.70095 2.245 Hours Worked .07380 .220 Constant . R2 = .25837 F-value - 4.84567 ** * =■ Significant at the 10% level. ** - Significant at the 5% level. 113 TABLE'XXIV English Equation For School Size 118-261 Without GPA Dependent Variable Independent Variable Regression . Coefficient F-Value ■ English School Attitude 2.55879 3.092 * Per Student Expend. - .00222 12.640 ** ■School Size - .08895 19.511 AA Father’s Education .26222 2.135 Items at Home 4.62512 .694 Reading Material .03720 1.391 Cars in Family - .52017 .865 Own Room and Car - .03288 .000 College Intention 4.65389 NEMCp -- School Activities 5.02088 4.177 AA Hours Worked - .00740 .022 . Constant 61.48145 R2 = .30156 F-value = 6.00541 *A * = Significant at the 10% level.. ** - Significant at the 5% level. 114 TABLE XXV Reading Equation For School Size 262-1040 Without GPA Dependent Independent Regression Variable Variable Coefficient F-Value Reading School Attitude 3.45234 10.825 ** Per Student Expend. .0079 .282 School Size .00697 7.531 ** Father's Education .16325 . 2.305 Items at Home 5.95594 1.657 Reading Material .10074 rrErNM -n Cars in Family .01519 .00 Ovm Room and Car -3.09590 eECMp -n College Intention 5.63838 . 31.122 ** School Activities 4.80936 CEYOM -n Hours Worked - .15155 18.568 ** Constant 32.67509 R2 = .30964 F-value = 15.24968 ** * = Significant at the 10% level ** = Significant at the 5% level 115 Numerical Competency Equation For School Size 262-1040 Without GPA TABLE XXVI Dependent Variable Independent Variable Regression Coefficient F-Value Numerical School Attitude 2.26896 5.101 *A Competency Per Student Expend. .00008 .003 School Size .00317 1.699 Father's Education .18644 3.279 * Items at Home 9.44222 4.542 ** Reading Material .07803 7.326 f f . Cars in Family .23664 .551 Own Room and Car -3.53542 6.596 ff College Intention 4.55640 22.170 f f School Activities 6.31275 12.880 *A Hours ■ Worked .00179 .003 Constant 33.37419 R2 = .24638 F-value = 11.11548 ** * = Significant at the 10% level. .** = Significant at the 5% level. I . 116 TABLE XXVII ' English Equation For School Size 262-1040 Without GPA Dependent Variable. Independent Variable Regression Coefficient F-Value English School Attitude 2.63059 7.207 ** ■Per Student Expend. .00157 1.268 School Size .00019 .007 'Father's Education .16017 2.544 Items at Home 4.99358 1.335 Reading Material .05655 4.045 ** Cars in Family - .16570 . 284 Own Room and Car -3.90038 8.439 ** .College Intention . 4.86124 26.529 Aft School Activities 5.63449 .. 10.787 ftft Hours Worked - .09178 7.809 ftft Constant 35.97157 R2 = .27406 F-value = 12.83561 ** * = Significant at the 10% level. ** = Significant at the 5% level. 117 ' Reading Equation For School Size 1040 and Greater Without GPA Dependent Independent Regression TABLE XXVIII Variable . Variable Coefficient . F-Value ■ Reading School Attitude - .3939,6 .064 Per Student Expend. - .00160 3.049 * . .School Size - .00031 .010 ‘ . Father's Education .52604 10.220 ** Items at Home 9.73697 .651 Reading Material - .10280 4.832 ** Cars in Family .13376 .055 Own Room and Car .59844 .072 ' College Intention 6.51965 18.548 ** School Activities 2.53765 .656 Hours Worked. . - .08439 2.959 * Constant 37.37697 R2 = .26237 F-value = 6.07897 ** * = Significant at the 10% level. ** = Significant at the 5% level. 118 Numerical Competency Equation For.School Size 1041 And Greater Without GPA TABLE XXIX Dependent Variable Independent Variable Regression Coefficient F-Value Numerical School Attitude .53126 .130 Competency Per Student Expend. - .0 0198 5.120 Aft School Size .0 0186 . .397 Father's Education .02375 .023 'Items at Home 1 7 .2 4 3 3 0 2 .2 7 9 Reading Material - .0 1137 .066 . Cars in Family .50527 ' i 874 Own Room and Car .7 7885 .136 College Intention 5.86738 , 16.749 Aft School Activities 3.01440 1.033 Hours Worked .00931. .040 Constant 29.12314 R2 = .1866475 F-value = 3.81936 Aft * = Significant at the 10% level. *A = Significant at the 5% level. 119 English Equation For School Size 1040 and Greater Without GPA TABLE XXX Dependent Variable Independent Variable Regression Coefficient F-Value English School Attitude -1.16831 .672 Per Student Expend. - .00151 3.263 * School Size .00079 .077 Father's Education .25149 2.781 * Items at Home 33.59801 9.241 *A Reading Material .00539 .016 Cars in Family - .73044 1.951 Own Room and Car -1.97778 .940 College Intention 4.48844 . 10.466 ** School Activities 3.68870 1.651 Hours Worked - .01171 .068 Constant 17.96560 R2 = .23459 F-value = 5.23812 ** * = Significant at the 10% level. ** - Significant at the' 5% level. APPENDIX A 121 Hleaee answer each question as correctly as possible. Bchool _____________________________________________ Student Mxunber . (from test sheet]" Who is the major provider In your familyT Father _____ Hother _____ Other _____ What Is the occupation of your fatherT What is the occupation of your motherT la your father self-employed! Yts __ *> Is your mother self-employed? Yes Ho Circle your father's highest educational level. I 2 3 Ii 5 6 7 6 9 10 11 12 13 Ik Ii 16 IT 18 19 20 Grade High School Ooll.f* Post-Grad Circle your mother's highest educational level. I 2 3 "I 5 6 T 8 9 10 11 12 13 Ik Ii 16 IT 18 19 20 Grade High School College Post-Grad Do you live at home with both of your parents? How many children are In your family, including yourself? Hov many cars are in your immediate family? ________ Circle those items you have in your home. telephone television newspaper radio What is the number of magasinee to which your family subscribes? Uo your parents purchase books? _____Yes __ to If yes, about how many a year for general family reading? ______ Do you have your own room? Yes __ *° Do you have your own car? Yes Mo 122 Circle the location that beet deecrlbee where you llvet Farm or Hanch In Town Out of Town Other How many hours a week do you spend on school studies outside of school hours? Uo you have a part time job (Include fare choree) during the school yesirt If yes, how many hours per week? ___________ Have you traveled: ____________ outside Montana ___________ outside the U.B. ___________ outside North America Yes Ho Does one of your parents attend parent-teacher conferences or school open-houses? Yes __ Mo Do you take part In: (Check If Yes) ______ Student Government _____ Athletics _____ Class Plays ______ Btudsnt Organisations _____ Other School Organisations What la your favorite subject? ____________________ Vliat do you plan to do when you graduate from high school? (Check one) Colleges (All) ____________ Vocational-Technical School Cosssunlty College Private Trade School ____________Armed forces ____________ Laployment Who Influenced you to do the above? (Check one) ____________ Another Student Teacher or Counselor ____________Mother ____________ Father Relative Other Do you like school? Yes No How long have you lived In this community? JKlB 3-21-73 APPENDIX B TAMLL XZXI Correlation Satria Him. School Per Student Beginning School Father's Iteme in Reading Family Own Room College School Hours Grade stiE Read. Attitude Expenditure Salary Size Education Material Antos and Car Intent. Activities Work. Point Total Score .889 .910 .860 .278 -.105 .041 -.025 .165 .079 .096 -.009 -.183 .425 .298 -.122 .643 BngliBh .742 .631 .237 -.077 .047 -.034 .143 .065 .098 -.066 -.124 .367 .286 -.132 .582 leading .667 .261 -.117 .020 -.014 .157 .063 .073 -.013 -.070 .395 .240 -.175 .572 Numerical Competency .240 -.083 .043 -.052 .136 .085 .087 .055 -.028 .367 .271 -.012 .557 School Attitude -.089 -.123 -.007 .076 -.018 .013 -.124 - 098 .290 .242 -.064 .286 Per Student Bicpenditure .215 -.236 -.036 -.045 .061 .035 .012 -.004 .101 .063 -.049 Beginning Salary .287 .056 .064 -.006 -.002 -.055 .035 -.117 -.062 .033 School Site .112 .116 -.101 -.061 .053 .036 -.408 .019 -.003 Father's Mucation .071 .125 .033 .056 .216 .087 .082 .122 It— in the Home -.035 111 .094 .079 .039 -.029 .148 loading Material .118 .105 .064 .115 .024 .032 Family Autos .294 -.040 -.007 .196 -.067 Cfen Room and Car -.045 -.006 .193 -.100 College Iatention .304 -.070 .400 School Activities -.044 .296 Hours Borked -.152 124 TAJLZ XXXI Correlation Satrlx Zn*. Read. Hum. School Attitude Per Student Expenditure Beginning Salary School Size Father's Education Items in Reading Material Family Autos Own Room and Car College Intent. School Activities Hours Work. Grade Point Total Score .889 .910 .860 .278 -.105 .041 -.025 .165 .079 .096 -.009 -.183 .425 .298 -.122 .643 English .742 .631 .237 -.077 .047 -.034 .143 .065 .098 -.066 -.124 .367 .286 -.132 .582 Reading .667 .261 -.117 .020 -.014 .157 .063 .073 -.013 -.070 .395 .240 -.175 .572 Numerical Competency .240 -.083 .043 -.052 .138 .085 .087 .055 -.028 .367 .271 -.012 .557 School Attitude -.089 -.123 -.007 .076 -.018 .013 -.124 -.098 .290 .242 -.064 .286 Per Student Expenditure .215 -.238 -.036 -.045 .061 .035 .012 -.004 .101 .063 -.049 Beginning Salary .287 .058 .084 -.006 -.002 -.055 .035 -.117 -.062 .033 School Sise .112 .116 -.101 -.061 .053 .036 -.408 .019 -.003 Father's Bdocatlon .071 .125 .033 .056 .216 .067 .082 .122 Iteem In the Borne -.035 .111 .094 .079 .039 -.029 .148 Reading Itoterlal .118 .105 .064 .115 .024 .032 Family Autoe .294 -.040 -.007 .196 -.067 Oen Room and Car -.045 -.006 .193 -.100 College Intention .304 -.070 .400 School Activities -.044 .296 Hours Borked -.152 125 T A T TTTT Correlation Matrix Ena. Read. Sun. Comp. School Attitude Per Student Eroenditure Beginning Salary School Size Father's Edocation Iteee in leading Material Family Autoe Own loon and Car College Intent. School Activities Work. Grade Point Total Score .889 .910 .860 .278 -.105 .041 -.025 .165 .079 .096 -.009 -.183 .425 .298 -.122 .643 Ragltsh .742 .631 .237 -.077 .047 -.034 .143 .065 .098 -.066 -.124 .367 .286 -.132 .582 leading .667 .261 -.117 .020 -.014 .157 .063 .073 -.013 -.070 .395 .240 -.175 .572 Competency .240 -.083 .043 -.052 .138 .085 .087 .055 -.028 .367 .271 -.012 .557 School Attitude -.089 -.123 -.007 .076 -.018 .013 -.124 -.098 .290 .242 -.064 .286 Per Student Expenditure .215 -.238 -.036 -.045 .061 .035 .012 -.004 .101 .063 -.049 leSlnnln* Smlmry .287 .058 .084 -.006 -.002 -.055 .035 -.117 -.062 .033 School Size .112 .116 -.101 -.061 .053 .036 -.408 .019 -.003 Father*• Edocation .071 .125 .033 .056 .216 .067 .082 .122 Iteea in the home -.035 .111 .094 .079 .039 -.029 .148 leading Material .118 .105 .064 .115 .024 .032 Faadly Antes .294 -.040 -.007 .196 -.067 One loon and Car -.045 -.006 .193 -.100 College Intention .304 -.070 .400 School Activities -.044 296 - . 1 5 2 126 BIBLIOGRAPHY 128 LITERATURE CITED Bowles, Samuel. Educational Production Function. Cambridge, Mass.: Harvard University Press, 1969. , arid Levin, Henry M. "The Determiriants of Scholastic Achieve­ ment: An appraisal of Some Recent Findings:11 Journal of Human Resources, III, No. I (Winter, 1968). ______, "More on Multicollinearity and the Effectiveness of Schools." Journal of Human Resources, III, No. 3 (Summer, 1968). Buros, Oscar Krisen, ed. The Seventh Mental Measurements Yearbook, Vols. I and II. Highland Park, N. J .: The Gryphon Press, 1972. Cochran, William G. Sampling Techniques. 2nd ed. New York: John Wiley and Sons, Inc., 1963. Coleman, James S. Equality of Educational Opportunity. U. S. Depart­ ment of Health, Education and Welfare, OE 38001. Washington, D.C.: Government Printing Office,1966. Doll, John P., V. James Rhodes and Jerry G. West. Economics of Agricultural Production, Markets and Policy. Homewood, 111.:. Richard D. Irwin;, Inc., 1968. Draper, N. R. and E . H. Smith. Applied Regression Analysis. New York: John Wiley and Sons. Inc., 1966. Ferguson, C. E. Microeconomic Theory. Homewood, 111,: Richard D. Irwin, Inc., 1972. Ferguson, George A. Statistical Analysis in Psychology arid Education. New York: McGraw-Hill Book Co., 1966. Gardner, Eric F., Jack C. Merwin, Robert Callis and Richard Madden. Stanford Achievement Test, High School BAttery Manual.. ■ NdW York: Harcourt, Brace and World, Inc., 1965. . . ' Guthrie, Harold W. Statistical Methods in Economics. Homewood, 111.: Richard D. Irwin, Inc,, 1966. Guthrie, James W. "A Survey of School Effectiveness Studies." in Do Teachers Make A Difference, U.S. Department of Health, Educ­ ation and Welfare. Washington, D.C.: Government Printing Office, 1970. ' 129 Klesllng, Herbert J . "Measuring a Local Government Service: A Study of School Districts in New York State," Review of Economics and Statistics, IL, No. 3 (August, 1967). Levin, Henry M. "A New Model of School Effectiveness," in Do Teachers Make A Difference. U. S. Department of .Health, Education and Welfare. Washington, D.C.: Government Printing Office, 1970. Lund, R. E. Multifactor Analysis of Variance Program. Bozeman, Mont.: Montana State University Computing Center, November, 1973. Malinvaud, E. Statistical Methods of Econometrics. Chicago: Rand ■ McNally and Co., 1966. Mayeske, George W. "Teacher Attributes and Social Achievement," in ■Do Teachers Make A Difference. U. S, Department of Health, Education and Welfare. Washington, D. C.: Government Printing Office, 1970. Montana, State Board of Equalization. 25th Biennisl Report, 1972. _______, Superintendent of Public Instruction. Montana Education Directory. Helena, Mont.: 1972. ________, Superintendent of Public Instruction. Montana School Stat­ istics . Helena, Mont.: 1972. Nie, Norman, Dale H. Bent, Hadlai Hull. Statistical Package for the Social Sciences. New York.: McGraw-Hill Book Co., 1970. Reiss, Albert'J.,Jr., Otis D. Duncan, Paul K. Hatt and Cecil C. North. Occupations and Social Status. New York: The Free Press of Glencoe, Inc., 1961. Riew, John. "Economics of Scale in High School Organization." The ■Review of Economics and Statistics. Boston: Department of Economics, Harvard University, August, 1966. Roscoe, John T . Fundamental Research Statistics for the Behavioral Sciences. New York: Holt, Rinehart and Winston, Inc., 1969. Schultz, Theodore W. Investment in Human Capital. New York: The Free Press, 1971. 130 Shelley, Herman Walker. "An Analysis of the Relationship Between Eight Factors and Three Measures of Quality in Thirty-nine South Carolina Secondary Schools." Unpublished Ph.D. Dissertation, University of Florida, 1957. . Schmedemann, Ivan W., Everett E. Peterson and Andrew H. Pine. Crisis in Public School Finance. Great Plains Agricultural Council. Pub. No. 59, 1972. Simpson, Robert James. "Selected Relationships Among Reported Expendi­ tures and Programs in Metropolitan School Districts," Disserta­ tion Abstracts. XXII, Ann Arbor, Mich.: Wayne State University, University Microfilms, 1961. Snedecor, George W. and William Cochran. Statistical Methods. 6th ed. Ames, Iowa: The Iowa University Press, 1967. Stinson, Thomas F . and Edward F . Krahmer. "Local School Expenditures and Educational Quality: A Correlation Analysis.V American Journal of Agricultural Economics, LI, No. 5 (December, 1969). Street, P., J. H. Powell and J . W. Hamblin. "Achievement of Students and Size of School." Journal of Educational Research, LV (1962). Sweeney, Thomas J . Rural Poor Students and Guidance. Boston, Mass.: Houghton Mifflin Co., 1971. Thomas, J . Alan. "Efficiency in Education: A Study of the Relation­ ship between Selected Inputs and Mean Test Scores in a Sample of Senior High Schools." Unpublished Ph.D. Dissertation, School of Education, Stanford. University, 1962. Treatway, John F. "A Study of Certain Factors Associated with the Quality of Education in Arkansas School Districts." Dissertation Abstracts. XXIII, Ann Arbor, Mich.: University of Arkansas, University Microfilms, 1962. U. S. Department of Agriculture,.Economic Research Service. "Rural People in the American Economy." Agr. Econ. Rpt. No. 101. Washington, D.. C.: Government Printing Office, 1966. U . S. Department of Commerce, Bureau of the Census. 1970 Census of Population, I, Part 28. Washington, D. C.: Government Printing Office, 1973. 131 U. S. Report of the President's National Advisory Commission on Rural Poverty. The People Left Behind. Washington, D. C.: Government Printing Office, 1967. Welch, Finis. "Measurement of the Quality of Schooling." American Economic Review, LVI, No. 2 (May, 1966). West, Jerry G. and Donald D. Osburn. ■ "Quality of Schooling in Rural •Areas." Southern Journal of Agricultural Economics, IV, No. I . (July, 1972). Zetler, Alan George. "Riding Time and School Size as Factors in the Achievement of Bus Transported Pupils.v Unpublished manuscript, Montana State University, 1970. '1CO MONTANA STATE UNIVERSITY LIBRARIES 'r&'if f