When a hypothesis about the mean of a population of unknown variance
is being tested, the test statistics z and t are given by precisely the same
formula; that is, for a given collection of sample data, both would have
the same value. If, in this exercise (N = 17), the value of t (or z) were
evaluated in terms of normal-curve sampling-error theory, how would the
probability of a Type I error compare in magnitude with that which would
obtain under the I-curve sampling theory?
15.7.14 Consider the speed reading experiment described in Section 9.1
of the text. Assume that 15 students, the experimental subjects, were
assigned to the speed reading course and 15 students served as control
subjects. Assume further that a reading comprehension test was administered to both groups with the following results:
178 ELEMENTARY STATISTICAL METHODS / STUDY MANUAL SOME SMALL-SAMPLE THEORY AND ITS APPLICATION 179 f j When a hypothesis about the mean of a population of unknown variance is being tested, the test statistics z and t are given by precisely the same formula; that is, for a given collection of sample data, both would have the same value. If, in this exercise (N = 17), the value of t (or z) were evaluated in terms of normal-curve sampling-error theory, how would the probability of a Type I error compare in magnitude with that which would obtain under the I-curve sampling theory? 15.7.14 Consider the speed reading experiment described in Section 9.1 of the text. Assume that 15 students, the experimental subjects, were assigned to the speed reading course and 15 students served as control subjects. Assume further that a reading comprehension test was admin- istered to both groups with the following results: Experimental Control 17E = 15 nc = 15 X E = 48 Xc = 46 Se = 5 Sc = 7 a If it is desirable to test the hypothesis that /Lc — pc = 0, what is the estimated standard error of the difference X E — Xc that is appropriate for use with f-distribution sampling theory? (Note: Express the answer in terms of the square root sign,) b What is the number of degrees of freedom of this estimate? c If the mean of the population represented by the experimental sample could be either greater or less than that of the population represented by the control sample, specify the appropriate critical region for a test of the hypothesis in question in terms of the scale of values of the test statistic I, given .05 as the level of significance. d What is the particular value oft for the given data? (Note: "ci-gc..r,c 2.3.) e What is the outcome of the test? If it is rejection, what alternative is indicated? (Express this outcome both symbolically and verbally.) 15.7.15 Suppose that in a situation similar to that of 13.4.8 the means and standard deviations reported in Table 13.A were based on samples of 8 analyzers and 12 nonanalyzers. a What is the estimated value of the standard error of the difference XA XNA that is appropriate for use with t-distribution sampling theory? (Note: Leave answer in terms of square root sign.) b What is the number of degrees of freedom of this estimate? c If .01 is used as the level of significance, specify the critical region appropriate to a test of the hypothesis of 13.4.8 in terms of the scale of values of the test statistic t. d What is the value of t for the given sample data? (Note: ay T,,,A 2.58.) e What is the outcome of the test? If it is rejection, what alternative is indicated? f Under what condition(s) is this test exact? 15.7.16 The developers of a new ninth-grade algebra course wished to compare its effectiveness with that of the traditional course in general use.' Forty-three teachers from schools throughout the United States agreed to participate in the experiment. Twenty-one of these were randomly selected to teach the new course while the other 22 continued teaching the tradi- tional course. At the end of the experimental year, students in all 43 class- rooms were given a test published by Educational Testing Service and a new test based on the principles incorporated into the new course. In this experiment the classroom is the sampling unit (see Section 11.3), and the mean score for each classroom is used as the classroom score, The means, variances, and standard deviations of these classroom scores are shown in Table 15.A. The situation described in this exercise is based on information given in chapter 28 of the Handbook of Multivariate Experimental Psychology, edited by R. B. Cattell (Rand McNally, Chicago, 1966). The particular set of data reported in Table I5.A was fabricated for purposes of illustration. However, these data do conform to the data given in Cattell. SOME SMALL-SAMPLE THEORY AND ITS APPLICATION 181 180 ELEMENTARY STATISTICAL METHODS / STUDY MANUAL Table 15.A New Course Traditional Course (nN = 21) (nT = 22) Traditional Mean = 12 Mean 18 Test S2 = 36 S2 = 25 S = 6 S 5 Mean = IS Mean = 12 New Test S2 = 16 52 = 4 5 = 4 S=2 Note: Parts a through e refer to the data of the traditional test. a Let PN represent the mean on the traditional test of a population of classes taking the new course and p T represent the mean on the traditional test of a population of classes taking the traditional course. What is the estimated standard error of the difference X T — XN that is appropriate for use with t-distribution sampling theory? (Note: Leave answer in terms of square root sign.) b What is the number of degrees of freedom of this estimate? c If a = .05 is used as the level of significance specify the critical region appropriate to a test of the hypothesis p T — FtN = 0 in terms of the scale of values of the 1-statistic. (Use a two-tailed critical region.) d What is the value of t for the given data for the traditional test? (Note: 1,72.) e What is the outcome of the statistical test? If it is rejection, what alternative is indicated? Note: Parts f through h refer to the data of the new test. f The standard error of the difference XT XN that is appropriate for use with t-distribution theory is approximately .98. Assume the experi- menters wish to test the hypothesis p i — ps = 0. What is the value of the t-statistic for the observed data? g If a = .05 (two-tailed), what is the outcome of the test? If it is rejection, what alternative is indicated? h On the basis of the data for the new test in Table 15.A, what condition necessary for an exact test of Na: P T — /IN = 0 is probably not met? i Given the results in Table 15.A and the outcomes of parts e and g, what conclusion seems reasonable? 15.7.17 Consider the situation of 13.8.23. Suppose it is necessary to test the hypothesis in question using only the data given for the first ten subjects in Table 13.D (exercise 13.8.24). a What is the estimated value of the standard error of the difference X 11-r2 - that that is appropriate for use with 1-distribution sampling theory? (Note: X 11 _ 12 - X 7 _ 8 = 1.6, SD = 2.42.) b What is the number of degrees of freedom of this estimate? c If .01 is used as the level of significance, specify the appropriate critical region in terms of the scale of values oft as a test statistic. d What is the particular value of I for the given data? e What is the outcome of the test? If it is rejection, what alternative is indicated? f Under what condition(s) is this test exact? 2 0 U 2 oz op oa- t) 15.9,18 The data from three experiments are shown below. For each situation decide whether t-distribution theory, large-sample normal theory, or some other procedure should be used to test Ho: p E = Experiment 1 E C XE = 17.1 Xc = 19.6 SE 2 = 5.3 Sc2 = 24.9 ri, = 28 = 25 Experiment 2 X E = 33.6 SE 2 = 6.3 n E = 14 Experiment 3 E XE = 100.3 SE 2 = 50.0 n E = 5 XC = 25.8 SC 2 = 8,9 nc = 14 C Xc = 90,5 Sc2 = 10.0 ne = 20 15.9.19 Consider Consider the situation described in 15.7.16. From Table 15.A, as it applies to the new test, it may be seen that the variance of the classes taking the new course is approximately four times that of the classes taking the traditional course. Assume these sample facts do, in fact, indicate that the populations are not equally variable with respect to the new test scores. In part g of 15,7.16, an a-level of .05 was selected to test Ho : — 11 N = O. a Is the true a-level exactly .05? b Consider the conclusions listed on page 359 of the text. Which of these best supports the use of the t-test of Ho: AT — PN = 0? tr Why Js it difficult to apply the conclusion selected in b to went 182 ELEMENTARY STATISTICAL METHODS / STUDY MANUAL SOM11 IMALL .SAMPLI 0 .10 Aril 14+ 15.9.20 A guidance counselor was interested lit aritartelaing "Whether group of students (E's) could be effectively motivated, through fridieldtill and group guidance activities, to achieve significantly higher rosulti On a standardized achievement test than another group of students (C's) who were not offered the same guidance services." 2 Regarding the hypothesis pE = pc, the investigator wrote: "The statistical technique used for com- paring the two groups was the t-test of differences between two means with separate group variance." a There were 135 students in each group, Accordina to the text, what procedure should have been used to test Ho: pE = pc? b Did the experimenter use the procedure you identified in a? Explain. 15.10.21 The mean of a random sample of ten scores is 40 and the standard deviation is 12. a Use t-distribution theory to establish the limits of the 95 percent confi- dence interval for the mean of the population represented by this sample. b Under what condition(s) is the confidence coefficient associated with the universe of intervals to which this interval belongs exactly .95? 15.10.22 Consider the problem of the elementary school supervisor as modified in 15.6.13 (i.e. X = 49 and S = 15 for a random sample of 17). a What are the limits of the 99 percent confidence interval for the mean