Question

A simple random sample of 100 postal employees is used to test if the average time postal employees have worked for the postal service has changed from the value of 7.5 years, as recorded 20 years ago. The sample of the current employees gave a mean service time of 7.9 years with a standard deviation of s = 2 years. Assume the distribution of the time the employees work in service jobs is approximately normal.

What are you to test?

Is the standard deviation equal to 2.

Is the mean equal to 2.

Is the variance equal to 20.

How long has 100 employees worked for the post-office.

Has the average employment changed from 7.5.

Answer 1

Here we need to test Has the average employment changed from 7.5 ?

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ = 7.57.5

Ha: μ ≠ 7.57.5

This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a two-tailed test is tc​=1.984.

The rejection region for this two-tailed test is R={t:∣t∣>1.984}

(3) Test Statistics

The t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that ∣t∣=2>tc​=1.984, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0482, and since p=0.0482<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 the population mean μ is different than 7.5, at the 0.05 significance level.