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 30% of the men and 24% 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 answer this question, we can perform a hypothesis test to determine if there is a significant difference in the proportion of men and women responding "yes" to the question.

Let's start by setting up the null and alternative hypotheses:

Null Hypothesis (H0): There is no difference in the proportion of men and women responding "yes" to the question.
Alternative Hypothesis (Ha): There is a difference in the proportion of men and women responding "yes" to the question.

In this case, we will use a two-proportion z-test to test the hypothesis. The formula for the test statistic (z) is:

z = (p1 - p2) / sqrt(p * (1 - p) * (1/n1 + 1/n2))

where:
p1 and p2 are the sample proportions of men and women responding "yes."
n1 and n2 are the sample sizes of men and women.
p is the pooled proportion calculated as (x1 + x2) / (n1 + n2), where x1 and x2 are the number of men and women responding "yes."

Now let's plug in the given values into the formula:

n1 = 200 (sample size for men)
n2 = 200 (sample size for women)
x1 = (0.30)(200) = 60 (number of men responding "yes")
x2 = (0.24)(200) = 48 (number of women responding "yes")

p1 = x1 / n1 = 60 / 200 = 0.30
p2 = x2 / n2 = 48 / 200 = 0.24

p = (x1 + x2) / (n1 + n2) = (60 + 48) / (200 + 200) = 108 / 400 = 0.27

Now, let's calculate the test statistic (z):

z = (0.30 - 0.24) / sqrt(0.27 * (1 - 0.27) * (1/200 + 1/200)) ≈ 1.55 (rounded to two decimal places)

The next step is to find the p-value associated with this test statistic. Since we want to test the difference, we will perform a two-tailed test.

Using a standard normal distribution table or a calculator, we find that the cumulative probability (p-value) for z = 1.55 is approximately 0.1203 (rounded to four decimal places).

Therefore, the p-value is 0.1203.

Now, to determine whether we can conclude that there is a significant difference at the 0.05 level of significance, we compare the p-value to the chosen significance level.

Since the p-value (0.1203) is greater than the significance level (0.05), we do not have enough evidence to reject the null hypothesis. Therefore, we cannot conclude that there is a significant difference in the proportion of men and women responding "yes" at the 0.05 level of significance.