Statistical Procedures
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o comoare pre-test and post-test performance within groups on the California Mathematics Test and the attitude scale, the difference between test scores was computed for each individual and a t-test applied. There was no reason to believe that the population sampled 43 +-i7i ~ 37 was normal nor that the population variances were equal. However, regarding the first assumption, Hays (1963, D- 322) states that ".. - this assumption (normality) may be violated with impunity providedthat sample size is not extremely small." Regarding the homogeniety ofvariance, Hays goes on to say ". . - for samples of equal size relativelybig differences in the population variances seem to have relativelysmall consequence for the eonclusions derived from a t-test." The t value obtained will be computed by the usual formula, t = dYn , whereSq d “>a, n and 84 = (n S$: dj - (5 43)2) n(n-1),Ti i=l i*1]with d; = posttest score minus pretest score for individual i.It was clear from the data gathered that individual differenceson the initial measures existed thus to control these initial differencesan analysis-of-covariance technique was used to test for a differenceof means of the various criterion measures.The analysis-of-covariance technique used compares the means oftwo samples using one or two associated independent variables (covariates). Analysis of covariance uses linear regression to predict criterion means based on the initial measures selected as covariates. Letting Y represent the criterion measure and X the covariate,the analysis is of the total sum of squares of the residuals, Y - Yx,where Y, = Y + by(X - X), with ¥ and X representing the sample means and bp the slope of the regression line of the total sample. In the case of two covariates, say X and Z, the residuals become yi «f¥ + Yue, 38where 5 ee = Y + byp (XK a * ba p(Z - Zz), with bh and bot representingthe regression coefficients Oy xs and Dynex for the entire group. Whenever the rejection region is "two-tailed," the F-ratio for the lower critical value will be found by the relationship ee |F (11m) LY where m and n represent the degrees of freedom (Hays, 1963, pe 350).The reader will recall that for a two-tailed test and a fixed significancelevel,x, the tables for F must be entered at «/2. The computational formulas and tests of hypothesis used arethose of Walker and Lev (1953, pp. 387-22). To insure that the covariates chosen were those that correlatedhighly with the criterion measure and also exhibited reasonable differ-ences, correlation coefficients between initial measures and criterionmeasures were computed. The correlation between initial measure andcriterion measure had to exceed 0.3 before consideration of the initialmeasure aS a covariate in the analysis of the means of the differentcriterion measureSse In addition to a comparison of the two groups in toto on thevarious criterion measures, the upper one-half ability levels and lowerone-half ability levels of each group as determined by the individual] A.C.T. mathematics percentile scores were compared.
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