Connie Rodrigues, the Dean of Students at Midstate

Answered question

2022-01-24

Connie Rodrigues, the Dean of Students at Midstate College, is wondering about differences in grades between Business school and College of arts & science. A quick random sampling produced the following GPAs (Grade Point Average). Business School: 2.86 2.77 3.18 2.8 3.14 2.87 3.19 3.24 2.91 3 2.83 Arts & Sciences 3.35 3.32 3.36 3.63 3.41 3.37 3.45 3.43 3.44 3.17 3.26 3.18 3.41 (a) Assuming that the GPAs are normally distributed, test whether there is difference in grades between Business School and Arts & Science College. Use α = 0.01. 07 (b) If the Dean believes that grades of Business School students are better than that of Arts & Science college, test her belief at α = 0.01. 07 OR Q.5 (a) Assuming that the GPAs are not normally distributed, apply appropriate non-parametric to test whether there is difference in grades between Business School and Arts & Science College. Use α = 0.01. 07 (b) Assuming that the GPAs are not normally distributed, apply appropriate non-parametric to test whether Business School students score better GPAs than students of Arts & Science College. Use α = 0.01

Answer & Explanation

user_27qwe

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 α=0.01.

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:
t=x1-x2s1n1+1n2
where x1 and x2 are the sample means, s is the pooled standard deviation, and n1 and n2 are the sample sizes.

Plugging in the values, we get:
t=2.98-3.320.218111+113=-3.42

The degrees of freedom for the test are df=n1+n2-2=22.

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:
t=2.98-3.320.218111+113=-3.42

The degrees of freedom for the test are df=n1+n2-2=22.

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 α=0.01.

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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