Question

The math department is trying to determine if their tests have a gender bias. They look at scores at old exams which were normally distributed and take a random sample of students. They found that the scores of the 16 male students had a normal distribution with a mean of 72.4 and a standard deviation of 7.1. The scores of the 17 females students had a normal distribution with a mean of 77.4 and a standard deviation of 5.2. Using a significance level of 0.05, test the claim the tests have a gender bias.

Answer 1

Objective: We have to test whether test scores have a gender bias.Let denote the mean scores of male and female students respecetively.

Given:

We have to test:

Vs . at  a significance level of 0.05

Since, here, we are comparing two independent groups (Males and Females), the appropriate statistical test to test the above hypothesis would be an independent sample test. Also, since the population standard deviations () are unknown we are left with the option of an independent sample t test.

Assuming that the following assumptions of the t test are satisfied:

- The data is normally distributed - The groups compared are independent - Homogenity of Variance

The test statistic is given by:

with Rejection region of the test given by

Substituting the given values:

= -0.374

Comparing the test statistic obtained with the critical value of t:  

Since, |t| = 0.374 < 2.039, we find that it does not lie in the rejection region. We fail to reject H0. We may conclude that our data does not provide sufficient evidence to support the claim that test scores have a gender bias.