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. David decides to carry out a formal test to validate his hunch regarding the difference in average scores. In a random sample of 18 students in the first section, he computes a mean and standard deviation of 77.4 and 10.8, respectively. In the second section, a random sample of 14 students results in a mean of 74.1 and a standard deviation of 12.2. a). Construct the null and the alternative hypotheses to test David’s hunch. b). Compute the value of test statistic. c). Conduct the test at α = 0.01 by critical value method and interpret your results.

Answer 1

a) Null hypothesis
`(\(H_0\))`: The average scores of students in the first and second sections of David Anderson's introductory accounting classes are equal. Alternative hypothesis
`(\(H_a\))`: There is a significant difference in the average scores between the two sections.

b) The test statistic for the hypothesis test is
`\(t = 1.97\)`.

c) At a significance level
`(\(\alpha\))` of 0.01, the critical value for a two-tailed test is
`\(\pm2.898\)`. Since
`\(|1.97| < 2.898\)`, we fail to reject the null hypothesis. There is not enough evidence to conclude that there is a significant difference in the average scores between the two sections.

Step-by-step explanation:

• To formulate the hypotheses, we set up the null hypothesis (\(H_0\)) as the assumption that there is no difference in the average scores between the two sections:
  `\(μ_1 = μ_2\)`. The alternative hypothesis
  `(\(H_a\))` suggests that there is a significant difference:
  `\(μ_1 \\eq μ_2\)`.

• To compute the test statistic, we use the formula
  `\(t = \frac{(\bar{X}_1 - \bar{X}_2)}{\sqrt{(S_1^2)/(n_1) + (S_2^2)/(n_2)}}\),` where
  `\(\bar{X}_1\)` and
  `\(\bar{X}_2\)` are the sample means,
  `\(S_1\)` and
  `\(S_2\)` are the sample standard deviations, and
  `\(n_1\)` and
  `\(n_2\)` are the sample sizes. Plugging in the values, we find
  `\(t = 1.97\)`.

• Using a critical value approach at a significance level
  `(\(\alpha\))` of 0.01 for a two-tailed test, the critical values are
  `\(\pm2.898\)` (obtained from a t-table). Since
  `\(|1.97| < 2.898\)`, we fail to reject the null hypothesis.

This implies that the observed difference in average scores is not statistically significant at the 0.01 level, and we do not have enough evidence to conclude that there is a significant disparity between the two sections.