Question

Consider the following results for two samples randomly taken from two populations.

Sample A Sample B

Sample Size 31 35

Sample Mean 106 102

Sample Standard Deviation 8 7

Test the hypothesis Ho=sigma1-sigma2=0 vs Ha=sigma1-sigma2 do not equal 0 at 5% level of significance. Show all six steps using p-value approach.

Answer 1

The provided sample means are shown below:

Also, the provided sample standard deviations are:

and the sample sizes are

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

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 = 0.05, and the degrees of freedom are df = 31+35-2 =64.

Hence, it is found that the critical value for this two-tailed test is t_c = 1.998.

The rejection region for this two-tailed test is R = {t: |t| > 1.998}

(3) Test Statistics

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

(4) Decision about the null hypothesis

Since it is observed that |t| = 2.167 > t_c = 1.998, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.034, and since p = 0.034 < 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 1​ is different than population mean 2 ​, at the 0.05 significance level.