The Adeles Workplace Insights Survey sampled men and women workers and asked if they expected to get a raise or promotion this year. Suppose the survey sampled 200 men and 200 women. If 104 of the men replied Yes and 74 of the women replied Yes, are the results statistically significant in that you can conclude a greater proportion of men are expecting to get a raise or a promotion this year?
a. State the hypothesis test in terms of the population proportion of men and the population proportion of women.
b. What is the sample proportion for men? For women?
c. Use a 0.01 level of significance. What is the p-value and what is your conclusion?
a. The hypothesis test in terms of the population proportion of men (p1) and the population proportion of women (p2) can be stated as follows:
Null hypothesis (H0): p1 = p2 (There is no difference between the proportions of men and women expecting to get a raise or promotion this year.)
Alternative hypothesis (Ha): p1 > p2 (A greater proportion of men are expecting to get a raise or promotion this year.)
b. To calculate the sample proportion for men and women, we divide the number of "Yes" responses by the size of the respective samples:
Sample proportion for men (P1) = 104 / 200 = 0.52
Sample proportion for women (P2) = 74 / 200 = 0.37
c. Using a 0.01 level of significance, we can calculate the p-value to determine the statistical significance of the results.
To calculate the p-value, we can use the two-sample proportion z-test. This test compares the difference in proportions (P1 - P2) to the null hypothesis value (0) and calculates the probability of obtaining a difference as extreme as the observed difference, assuming the null hypothesis is true.
The formula to calculate the test statistic (z) is:
z = (P1 - P2) / sqrt( (P_total * (1 - P_total)) * ( (1/N1) + (1/N2) ) )
Where P_total = (x1 + x2) / (n1 + n2), N1 and N2 are the respective sample sizes, and x1 and x2 are the number of "Yes" responses for men and women, respectively.
Using the given values, we can calculate the test statistic:
z = (0.52 - 0.37) / sqrt( ( (104 + 74) / (200 + 200) ) * ( (1/200) + (1/200) ) )
= 1.40
Next, we find the p-value associated with the test statistic using a standard normal distribution table or a statistical software.
The p-value represents the probability of obtaining a test statistic as extreme as the observed one, assuming the null hypothesis is true. Since we are conducting a one-sided test (p1 > p2), we are interested in the area to the right of the test statistic.
Assuming the calculated p-value is less than the significance level (0.01), we reject the null hypothesis. If it is greater than the significance level, we fail to reject the null hypothesis.
Hence, the p-value and the conclusion depend on the specific value obtained from the standard normal distribution table or statistical software.