Question

Salaries of men and women
Woman	Man
26	29
28	31
30	33
32	29
34	33
48	56
52	54
22	28
27	33

You are trying to determine whether male and female Central Bank employees, having equal qualifications, receive different salaries. The data contain the salaries (in thousands of dollars) for 9 male and 9 female employees. Assume salaries are normally distributed.

a. Assume that each row of data represents paired observations, and using alpha=0.05, can you conclude that members of different genders are paid equally? Be sure to write down your hypothesis.

b. How would you collect data to ensure that the observations are actually paired?

Answer 1

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μD​ = 0

Ha: μD​ ≠ 0

This corresponds to a two-tailed test, for which a t-test for two paired samples be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df=8.

Hence, it is found that the critical value for this two-tailed test is tc​=2.306, for α=0.05 and df=8.

The rejection region for this two-tailed test is R={t:∣t∣>2.306}.

(3) Test Statistics

The t-statistic is computed as shown in the following formula:

t=​D¯​/sD​/sqrt(n)

t=3.464/9​−3​

t=−2.598

(4) Decision about the null hypothesis

Since it is observed that ∣t∣=2.598>tc​=2.306, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0317, and since p=0.0317<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion: It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean μ1​ is different than μ2​, at the 0.05 significance level.

Therefore we can conclude that members of different genders are not paid equally.

b) We would collect the data like Woman and Man both hold the same degrees like both are MBAs or both have same amount of working experience.