Question

The number of men and women among professors in Math, Physics, Chemistry, Linguistics, and English departments from a SRS of small colleges were counted, and the results are shown in the table below. Dept. Math Physics Chemistry Linguistics English Men 60 98 36 29 38 Women 8 9 8 17 17 Test the claim that the gender of a professor is independent of the department. Use the significance level α=0.025

A) The test statistic is x² = ?
B) The critical value is x² = ?

Answer 1

To test if gender of a professor is independent of the department, we perform a Chi-square test of independence by comparing observed frequencies to expected frequencies. The test statistic and critical value would be calculated and compared at the given significance level to either reject or fail to reject the null hypothesis.

Step-by-step explanation:

To test the claim that the gender of a professor is independent of the department, we would perform a Chi-square test of independence. This statistical test will help us determine if there is a statistically significant association between two categorical variables, in this case, gender and department. The process involves observing the frequencies in each category and comparing them to the frequencies we would expect if there were no association between the variables.

Steps to Perform the Chi-square Test:

• Calculate the expected frequencies for each cell in the table assuming independence.
• Compute the Chi-square test statistic, which is the sum of the squared differences between observed and expected frequencies, divided by the expected frequencies.
• Compare the calculated Chi-square statistic to the critical value from the Chi-square distribution with the appropriate degrees of freedom to determine if the observed values significantly differ from the expected values. The degrees of freedom for this test is (number of rows - 1) * (number of columns - 1).
• Determine whether to reject or fail to reject the null hypothesis based on whether the Chi-square statistic is greater than the critical value at the given significance level.

We do not have the actual data to calculate the test statistic or the critical value here. Normally, this would be done using a statistical software or tables after calculating the expected frequencies for each cell. Once the test statistic is calculated, it would be compared to the critical value obtained from Chi-square distribution tables corresponding to the requisite degrees of freedom and the specified significance level.