Help Me Please!

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 28% 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.)

(b) State the appropriate conclusion.

Reject the null hypothesis, there is significant evidence that the proportions differ.
Reject the null hypothesis, there is not significant evidence that the proportions differ.
Fail to reject the null hypothesis, there is not significant evidence that the proportions differ.
Fail to reject the null hypothesis, there is significant evidence that the proportions differ.

H0: p1 = p2

Ha: p1 < p2

P1 = .28
p2 = .24

x1 = 200(.28) = 56
x2 = 200(.24) = 48
n1 = 200
n2 =200
z = -0.91
p-value = 0.1809

b. fail to reject the null hypothesis, there is not sufficient evidence that

To find the answer to this question, we can use hypothesis testing for the difference in proportions.

(a) First, we need to find z, which is calculated using the formula:

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

Where:
p1 = proportion of men responding "yes"
p2 = proportion of women responding "yes"
n1 = sample size for men
n2 = sample size for women

Given:
p1 = 28% = 0.28
p2 = 24% = 0.24
n1 = 200
n2 = 200

Substituting these values into the formula, we can calculate z:

z = (0.28 - 0.24) / sqrt((0.28 * (1 - 0.28) / 200) + (0.24 * (1 - 0.24) / 200))

Calculating this expression, we find z ≈ 1.32 (rounded to two decimal places).

(ii) To find the p-value, we need to determine the probability of obtaining a test statistic as extreme as our calculated z value (or more extreme) assuming the null hypothesis is true. Since the alternative hypothesis is two-sided, we will calculate the p-value as double the area in the tails beyond z.

Using a standard normal distribution table or a statistical software, we can find the area beyond 1.32 in the tails to be approximately 0.0922.

Therefore, the p-value ≈ 2 * 0.0922 ≈ 0.1844 (rounded to four decimal places).

(b) To state the appropriate conclusion, we compare the p-value to the significance level (α = 0.05).

Since the p-value (0.1844) is greater than the significance level (0.05), we fail to reject the null hypothesis. This means we do not have significant evidence to conclude that there is a difference in the proportion of men and women responding "yes" at the 0.05 level of significance.

Therefore, the correct answer is: Fail to reject the null hypothesis, there is not significant evidence that the proportions differ.