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EPS 525 – INTRODUCTION TO STATISTICS INDEPENDENT-SAMPLES t TEST – PRACTICE EXERCISE – KEY Dr. Kureous, a teacher at George Junior High, wants to determine if there is a significant difference between the 6th grade boys and girls in his class on their spelling exam. He does not have a prediction as to whether the boys or the girls will be better – he simply wants to see if they differ significantly on the exam. 1. State (using symbols and words) the null hypothesis and the alternative hypothesis for this independent-samples t test. H0: µ1 = µ2 or µ1 – µ2 = 0 There is no (statistically significant) difference between the boys’ spelling exam mean (µ1) and the girls’ spelling exam mean (µ2). • Ha: Could also make reference to the hypothesized mean difference = 0. µ1 ≠ µ2 or µ1 – µ2 ≠ 0 There is a (statistically significant) difference between the boys’ spelling exam mean (µ1) and the girls’ spelling exam mean (µ2). • Could also make reference to the hypothesized mean difference ≠ 0. 2. Has the assumption of independence been met for this data? YES or NO (circle your answer selection) Justify/explain your answer – that is, how did you come to your conclusion? The two groups (boys and girls) are independent of each other. 3. Has the assumption of normality been met for this data? Use an alpha (α) level of .001 looking at the Shapiro-Wilks test to justify your answer. YES or NO (circle your answer selection) Justify/explain your answer – that is, how did you come to your conclusion? (Hint: look at the SPSS printout). Be sure to show all applicable values and symbols. Looking at the p (Sig.) values on the Shapiro-Wilks test, we find that neither group/level was significant, thereby indicating that each of the levels of the independent variable (gender) are statistically normal (normally distributed). Boys: p (.818) > α (.001) Girls: p (.464) > α (.001) 4. Has the assumption of homogeneity of variance been met for this data? Use an alpha (α) level of .05 looking at the Levene’s Test for Equality of Variances. YES or NO (circle your answer selection) Justify/explain your answer – that is, how did you come to your conclusion? (Hint: look at the SPSS printout). Be sure to show all applicable values and symbols. The Levene’s Test for Equality of Variances showed: F = .807, p = .377 Compared to α = .05, p (.377) > α (.05) – retain the null hypothesis of no difference. Therefore it is not significant and the assumption is met – that is, equal variances are assumed. 5. After computing the test statistic (t-test) – complete the following tables for the independentsamples t test (Do not round – use the values from SPSS): Gender N Mean Standard Deviation Standard Error of the Mean Boys 15 74.87 6.791 1.754 Girls 15 81.40 9.485 2.449 t df Sig. (2-tailed) Mean Difference -2.169 28 .039 -6.53 95% CI of the Difference Lower Upper -12.703 -.363 6. Using an alpha (α) level of .05, interpret the results from the independent-samples t test: 6.a. Did you reject the null hypothesis for the group means in favor of the alternative hypothesis (indicated in question 1)? YES 6.b. or NO (circle your answer selection) Justify your answer, that is, how did you come to your conclusion? DO NOT make reference to the t critical value or the confidence intervals. Be sure to include the applicable values and symbols. t (28) = -2.169, p = .039 Compared to α = .05, p (.039) < α (.05) – which is significant, therefore the null hypothesis of no difference is rejected. INDEPENDENT-SAMPLES t TEST – KEY PAGE – 2 If applicable (or indicate otherwise, and why), calculate the Effect Size – be sure to show your work. d =t N1 + N 2 15 + 15 30 = −2.169 = −2.169 = −2.169 .133333 = −2.169(.365148) N1 N 2 (15)(15) 225 d = .7920068 = .79σ σ 5.c. Briefly discuss your findings (i.e., what do the results indicate or mean). Be sure to make reference to the group means and effect size (if applicable). Be sure to show all applicable values and symbols including the statistical strand for the results. The girls (M = 81.40, SD = 9.49) performed significantly better (higher) than the boys (M = 74.87, SD = 6.79) on the spelling test for this sample at the .05 alpha level, t(28) = 2.17, p < .05, d = .79. The mean difference (6.53) was significantly different from zero (0), with an effect size of just over threefourths (d = .79) of a standard deviation. Note: there are several ways to interpret the results, the key is to indicate that there was a significant difference between the boys and girls on the spelling test at the .05 alpha level – and include, at a minimum, reference to the group means and effect size (if applicable). t(28) = 2.17, p < .05, d = .79 t Indicates that we are using a t-Test (28) Indicates the degrees of freedom associated with this t-Test 2.17 Indicates the obtained t statistic value (tobt) p < .05 Indicates the probability of obtaining the given t value by chance alone d = .79 Indicates the effect size for the significant effect (measured in standard deviation units) INDEPENDENT-SAMPLES t TEST – KEY PAGE – 3 EPS 525 – INTRODUCTION TO STATISTICS INDEPENDENT-SAMPLES t TEST – PRACTICE EXERCISE – KEY Tests of Normality a Spelling Gender Boys Girls Kolmogorov-Smirnov Statistic df Sig. .130 15 .200* .183 15 .187 Statistic .967 .946 Shapiro-Wilk df 15 15 Sig. .818 .464 *. This is a lower bound of the true significance. a. Lilliefors Significance Correction Independent T-Test Group Statistics SPELLING GENDER 1 Boys 2 Girls N Mean 74.87 81.40 15 15 Std. Deviation 6.791 9.485 Std. Error Mean 1.754 2.449 Independent Samples Test Levene' s Test for Equality of Variances SPELLING Equal variances assumed Equal variances not assumed F .807 Sig. .377 t-test for Equality of Means t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper -2.169 28 .039 -6.53 3.012 -12.703 -.363 -2.169 25.367 .040 -6.53 3.012 -12.732 -.334