Question

A local shoe company claims that the average yearly salary of its workers is $19,800. The union believes that the average salary is much less. A random sample of 64 employees shows their average salary to be $19,400 with a sample standard deviation of $800. Use a 5% (alpha = 0.05) level of significance to test the company’s claim. Provide statistical evidence to support your findings.

Answer 1

The sample mean is and the sample standard deviation is s = 800, and the sample size is n = 64.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ = 19800. Average yearly salary of its workers is $19,800

Ha: μ < 19800. Average yearly salary of its workers is less than $19,800

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

(2) Rejection Region

The significance level is α=0.05, and the critical value for a left-tailed test is .

The rejection region for this left-tailed test is R = {t : t < − 1.669}

(3) Test Statistics

The t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that t = −4 < tc ​= −1.669, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0001, and since p = 0.0001 < 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 less than 19800, at the 0.05 significance level.

Therefore, there is enough evidence to claim that the average salary is much less.

Confidence Interval

The 95% confidence interval is 19200.166 < μ < 19599.834.