Question

A sample of employees in a company with executives and non-executives are asked whether they are satisfied with their present salary. TABLE A gives the summary of results: TABLE A Employees Sample Size Proportion of employees satisfied with present salary 18 50 Executives Non- executives 50 14 i. Test the hypothesis that the percentage of employees who are satisfied with their salaries is higher for executives compared to non-executives at 5% significance level. Determine the p-value for the test and state your conclusion.

Answer 1

Sample 1 be Executives

Sample 2 be non executives

For sample 1, we have that the sample size is N_1= 50, the number of favorable cases is X_1 = 18, so then the sample proportion is

X1 18 N = 5 = 0.36

For sample 2, we have that the sample size is N_2 = 50, the number of favorable cases is X_2 = 14, so then the sample proportion is $\hat p_2 = \frac{X_2}{N_2} = \frac{ 14}{ 50} = 0.28$

The value of the pooled proportion is computed as $\bar p = \frac{ X_1+X_2}{N_1+N_2}= \frac{ 18 + 14}{ 50+50} = 0.32$   

Also, the given significance level is α=0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho::p1​=p2​

Ha: p1​>p2​

This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a right-tailed test is z_c = 1.64

(3) Test Statistics

The z-statistic is computed as follows:

$z = \frac{\hat p_1 - \hat p_2}{\sqrt{ \bar p(1-\bar p)(1/n_1 + 1/n_2)}} = \frac{ 0.36 - 0.28}{\sqrt{ 0.32\cdot(1-0.32)(1/50 + 1/50)}} = 0.857$

(4) Decision about the null hypothesis

Since it is observed that z = 0.857 < z_c = 1.64, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.1956, and since p = 0.1956 > 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 proportion p1​ is greater than p2​, at the 0.05 significance level.