Question

A researcher wants to determine if the average salaries for professors at all public and all private universities are different. He randomly selects 30 professors from public universities and independently 30 professors from private universities. The data are in thousands of dollars. The statistics are given below. Test the claim that the average salaries for all professors at public and private universities are different. Use α = 0.05 and fully justify!

Mean	65.5	76.8	-11.3
Standard Deviation	24.3	24.3	24.3
Observations	30	30	30

Answer 1

The provided sample means are shown below:

Also, the provided sample standard deviations are:

and the sample sizes are n1​=30 and n2​=30.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​ = μ2​

Ha: μ1​ ≠ μ2​

This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df = 58 . In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal:

Hence, it is found that the critical value for this two-tailed test is t_c = 2.002 , for α=0.05 and df = 58

(3) Test Statistics

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:

t = -1.801

(4) Decision about the null hypothesis

Since it is observed that |t| = 1.801 < tc​=2.002, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.0769 , and since p = 0.0769 ≥0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1​ is different than μ2​, at the 0.05 significance level.