2022-01-24
user_27qwe
Skilled2023-04-22Added 375 answers
Let's solve the given math problem step by step:
(a) Assuming that the GPAs are normally distributed, we can use a two-sample t-test to test whether there is a difference in grades between Business School and Arts & Science College. Here, our null hypothesis is that the mean GPAs of the two schools are equal, while the alternative hypothesis is that they are not equal. We use a significance level of .
Using statistical software or a calculator, we find that the sample mean GPA of Business School is 2.98 and the sample mean GPA of Arts & Science College is 3.35. The pooled standard deviation is 0.218.
The t-statistic is given by:
where are the sample means, s is the pooled standard deviation, and are the sample sizes.
Plugging in the values, we get:
The degrees of freedom for the test are .
Using a t-distribution table or software, we find that the p-value for the test is less than 0.01. Therefore, we reject the null hypothesis and conclude that there is a statistically significant difference in GPAs between Business School and Arts & Science College.
(b) To test the Dean's belief that grades of Business School students are better than that of Arts & Science College, we need to conduct a one-tailed t-test with the null hypothesis that the mean GPA of Business School is less than or equal to that of Arts & Science College, and the alternative hypothesis that the mean GPA of Business School is greater than that of Arts & Science College. We again use a significance level of α = 0.01.
Using the same formula for the t-statistic and plugging in the values, we get:
The degrees of freedom for the test are .
Using a t-distribution table or software, we find that the p-value for the test is less than 0.005 (half of the two-tailed p-value). Therefore, we reject the null hypothesis and conclude that there is evidence to support the Dean's belief that grades of Business School students are better than that of Arts & Science College at a significance level of .
Alternatively, we could have used a one-tailed test with the null hypothesis that the mean GPA of Arts & Science College is less than or equal to that of Business School, and the alternative hypothesis that the mean GPA of Arts & Science College is greater than that of Business School. The results would be the same.
(c) If the GPAs are not normally distributed, we can use the Wilcoxon rank-sum test (also known as the Mann-Whitney U test) to test for a difference in grades between Business School and Arts & Science College. This test does not assume normality and is appropriate for comparing two independent samples. We use a significance level of α = 0.01.
The null hypothesis for the test is that the distribution of GPAs is the same for the two schools, while the alternative hypothesis is that the distribution of GPAs
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