In a survey of families in which both parents work, one of the questions asked was, "Have you refused a job, promotion, or transfer because it would mean less time with your family?" A total of 200 men and 200 women were asked this question. "Yes" was the response given by 33% of the men and 30% of the women. Based on this survey, can we conclude that there is a difference in the proportion of men and women responding "yes" at the 0.05 level of significance? (Use Men - Women.)

(a) Find z. (Give your answer correct to two decimal places.)


(ii) Find the p-value. (Give your answer correct to four decimal places.)

To determine if there is a difference in the proportion of men and women responding "yes", we can conduct a two-sample z-test.

First, calculate the sample proportions for men and women who responded "yes":
Proportion for men: 33% = 0.33
Proportion for women: 30% = 0.30

Next, we need to calculate the standard error for the difference in proportions:
Standard error = sqrt[(p1(1-p1)/n1) + (p2(1-p2)/n2)]
where:
p1 = proportion for men
p2 = proportion for women
n1 = sample size for men
n2 = sample size for women

Substituting the values, we get:
Standard error = sqrt[(0.33(1-0.33)/200) + (0.30(1-0.30)/200)]

Calculating this expression:
Standard error ≈ 0.0321

Now, we can calculate the z-score:
z = (p1 - p2) / standard error
where p1 - p2 is the difference in proportions

Substituting the values, we get:
z = (0.33 - 0.30) / 0.0321

Calculating this expression:
z ≈ 0.0935

(i) The z-score is approximately 0.09.

To find the p-value, we need to consult a z-table or use appropriate software/statistical calculator.
From the z-table, we can find that the cumulative proportion from 0 to 0.09 is approximately 0.5375.

Since we are conducting a two-tailed test, we double the cumulative proportion:
p-value ≈ 2 * 0.5375

Calculating this expression:
p-value ≈ 1.0750

(ii) The p-value is approximately 1.0750.

To make a conclusion, we compare the p-value with the significance level of 0.05.
Since the p-value (1.0750) is greater than the significance level (0.05), we fail to reject the null hypothesis.

Therefore, based on this survey, we do not have enough evidence to conclude that there is a difference in the proportion of men and women responding "yes" at the 0.05 level of significance.