Question

David Anderson has been working as a lecturer at Michigan State University for the last three years. He teaches two large sections of introductory accounting every semester. While he uses the same lecture notes in both sections, his students in the first section outperform those in the second section. He believes that students in the first section not only tend to get higher scores, they also tend to have lower variability in scores. David decides to carry out a formal test to validate his hunch regarding the difference in average scores. In a random sample of 20 students in the first section, he computes a mean and a standard deviation of 85.2 and 25.9, respectively. In the second section, a random sample of 21 students results in a mean of 85.0 and a standard deviation of 1.18.

Sample 1 consists of students in the first section and Sample 2 represents students in the second section.

a.	Construct the null and the alternative hypotheses to test David’s hunch.
	H0: μ1 − μ2 = 0; HA: μ1 − μ2 ≠ 0 H0: μ1 − μ2 ≥ 0; HA: μ1 − μ2 < 0 H0: μ1 − μ2 ≤ 0; HA: μ1 − μ2 > 0

b-1.	Calculate the value of the test statistic. (Round all intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)

Test statistic

b-2.	What assumption regarding the population variances is used to conduct the test?
	Known population standard deviations. Unknown population standard deviations that are equal Unknown population standard deviations that are not equal

c.	Implement the test at α = 0.10 using the critical value approach.
	Reject H0; there is evidence that scores are higher in the first section. Reject H0; there is no evidence that scores are higher in the first section. Do not reject H0; there is evidence that scores are higher in the first section. Do not reject H0; there is no evidence that scores are higher in the first section.

Answer 1

Let   denote the mean score of sample of students and their standard deviations in first and second section respectively. Let denote the population mean score of students in first and second sections respectively.

We have to test whether mean score of first section is greater than the second section. The negation of the claim to be tested would be the null hypothesis and the claim as such would be the alternative hypothesis.

Hence, we would test:

a. Vs

The appropriate statistical test here, would be an independent sample t test for means. To run the above test, we need to test whether the following assumptions are satisfied:

1.All the observations are independent 2.The dependent variable (here, scores of students) is continuous 3.The dependent variable is normally distributed 4. Homogeneity of variance

To check the assumption of homogeneity of variance, we may also test the claim whether scores of students in first section has lesser variability than that of the second.

To test: Vs   

Using the test statistic:

with left tail critical region being F < 0.464

  

= 481.7653

Comparing the test statistic with the critical F value :

Since, F = 481.7653 > 0.464 does not lie in the rejection region, we fail to reject H₀. We may conclude that we do not have sufficient evidence to support David's claim that students in the first section tend to have lower variability in scores.

The above test also implies that the assumption of homogeneity of variance for t test is satisfied.

Hence, b-2. Unknown population standard deviations that are equal

Assuming the data is normally distributed, the t test statistic is given by:

with critical region being t > t_critical

b-1. t = 0.035

Comparing the above test statistic with the critical value for n₁ + n₂ - 2 = 20 + 21 - 2 = 39 degrees of freedom at say, 5% level of significance for one tailed test:

Using excel function: TINV(0.10, 39) = 1.685

Since, t = 0.035 < 1.685 does not lie in the rejection region, we fail reject H₀ at 5% level. We may conclude that we do not have sufficient evidence to support David's claim that  students in the first section tend to get higher scores.

c. Do not reject H₀; there is no evidence that scores are higher in the first section.