Question

A workshop is designed to help students prepare for college algebra, and we want to evaluate its effectiveness. There are 10 college students with similar math skills selected in the study. The student take a test before they attend the workshop, then after they attend the workshop, they will take another test with the similar difficulty level. The data are collected and summarized as below. Perform an appropriate statistical inference to answer the research questions. Mean Standard Deviation Min Max Before workshop 75 6.2 68 84 After Workshop 83 7.2 72 88 Diff - After - Before 6 5.8 - 1 12 (Please be aware that there might be more information than needed) (1) Use appropriate inference procedure to analyze the data. (if it is two independent sample t test, use pooled variance t test) (2) Based on your inference, what type of error could you make? Which of the following best describe the error you indicated? A. The workshop is actually effective, but we concluded it is not. B. The workshop is actually not effective, but we concluded it is.

Answer 1

We'll be testing whether mean score $\mu$ ₁ < $\mu$ ₂ i.e mean score before workshop is less than that after workshop

The provided sample means are shown below: X1 = 75 X2 = 83 Also, the provided sample standard deviations are: S1 = 6.2 S2 = 7.2 and the sample sizes are ni = 10 and n2 = 10. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: Mi = 12 Ha: Mi < 12 This corresponds to a left-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used. (2) Rejection Region Based on the information provided, the significance level is a = 0.05, and the degrees of freedom are df = 18. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal:


Hence, it is found that the critical value for this left-tailed test is te = -1.734, for a = 0.05 and df 18. The rejection region for this left-tailed test is R = {t:t< -1.734}. (3) Test Statistics Since it is assumed that the population variances are equal, the t-statistic is computed as follows: t= X1 - X2 (ni-1)si+(n2 -1)s ni+ng-2 1 ni + n2 = 75 – 83 (10-1)6.22+(10-1)7.22 Go + 1) 10+10-2 -2.663 (4) Decision about the null hypothesis Since it is observed that t = -2.663 <tc = -1.734, it is then concluded that the null hypothesis is rejected. Using the P-value approach: The p-value is p= 0.0079, and since p=0.0079 < 0.05, it is concluded that the null hypothesis is rejected. (5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean Mi is less than um, at the 0.05 significance level.

2)
option B.