Question

1. In a study examining whether taking a statistics course influences students’ reasoning skill, a researcher first gave a reasoning task to 5 students who have never taken a statistics course, and then the students took a brief statistics lecture which lasted 3 hours. After the lecture, the same five students took the reasoning task again. Their performance scores are summarized in the table below.

Student ID	Performance before the lecture	Performance after the lecture
A	6	8
B	7	7
C	5	7
D	8	8
E	7	10

Perform an appropriate statistical test and provide your conclusion on whether or not their reasoning performance are different before and after taking the STATS lecture.

2. In the above (#9), the same 5 students took the test twice, before and after the lecture. Reanalyze the above data as the data obtained from two sets of independent samples. Now please perform an appropriate statistical test to compare the means, pretending that two different (independent) groups of five people took the same test, one before and one after the lecture.

Answer 1

Here first we will use paired t test to  compare the means and then we will use independent sample t-test to compare mean

so first we use paired t test

Therefore, there is not enough evidence to claim that the population mean of two groups are different at the 0.05 significance level.

2)

Reanalyze the above data as the data obtained from two sets of independent samples. So we will use independent sample t-test

Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean ​ are different for two independent groups ​, at the 0.05 significance level.