Question

A survey of 800 adults from a certain region​ asked, "What do you buy from your mobile​ device?" The results indicated that 57​% of the females and 48​% of the males answered clothes. The sample sizes of males and females were not provided. Suppose that of 500 ​females, 285 reported they buy clothing from their mobile​ device, while of 300 ​males, 144 reported they buy clothing from their mobile device.

A. is there evidence of a difference between males and females in the proportion who said they buy clothing from their mobile device at the 0.05 level of​ significance?

The null and alternative​ hypotheses is:

H0​: π1=π2

H1​: π1≠π2

Determine the value of the test statistic

ZSTAT=

Determine the critical​ value(s) for this test of hypothesis=

State the conclusion:

B. Find the​ p-value in​ (a) and interpret its meaning:

C. Construct and interpret a 90%, 95%, and 99​% confidence interval estimate for the difference between the proportion of males and females who said they buy clothing from their mobile device.

D: What are your answers to​ (a) through​ (c) if 432 males said they buy clothing from their mobile​ device?

Determine the value of the test statistic

Determine the value of X2=

Determine the proportion of items of interest in sample​ 2, p2=

ZSTAT=

The critical values for this test of hypothesis =

​p-value=

Construct and interpret a90%, 95%, and 99​% confidence interval estimate for the difference between the proportion of males and females who said they buy clothing from their mobile device.

Answer 1

Solution:-

A)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis:
Alternative hypothesis:

Note that these hypotheses constitute a two-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a two-proportion z-test.

Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).

p = 0.53625

SE = 0.03642
z = (p₁ - p₂) / SE

z = 2.47

z_Critical= + 1.96

Rejection region is - 1.96 > z > 1.96

where p₁ is the sample proportion in sample 1, where p₂ is the sample proportion in sample 2, n₁ is the size of sample 1, and n₂ is the size of sample 2.

Interpret results. Since the z-value (2.47) lies in the rejection region, hence we have to reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that there is evidence of a difference between males and females in the proportion who said they buy clothing from their mobile device at the 0.05 level of​ significance.

B)

Since we have a two-tailed test, the P-value is the probability that the z-score is less than -2.47 or greater than 2.47.

P-value = P(z < - 2.47) + P(z > 2.47)

Use z-calculator to find the p-values.

P-value = 0.007 + 0.007

Thus, the P-value = 0.014

Interpret results. Since the P-value (0.014) is less than the significance level (0.05), we cannot accept the null hypothesis.

From the above test there is evidence of a difference between males and females in the proportion who said they buy clothing from their mobile device at the 0.05 level of​ significance.

C)

90% confidence interval estimate for the difference between the proportion of males and females who said they buy clothing from their mobile device is C.I = (0.0301, 0.1499)

The 90% confidence interval contains all the values greater than 0, hence there is evidence of a difference between males and females in the proportion who said they buy clothing from their mobile device at the 0.05 level of​ significance.

95% confidence interval estimate for the difference between the proportion of males and females who said they buy clothing from their mobile device is C.I = (0.0186, 0.1614)

The 95% confidence interval contains all the values greater than 0, hence there is evidence of a difference between males and females in the proportion who said they buy clothing from their mobile device at the 0.05 level of​ significance.

99​% confidence interval estimate for the difference between the proportion of males and females who said they buy clothing from their mobile device is C.I = (-0.0038, 0.1838).

The 99% confidence interval contains 0, hence there is not evidence of a difference between males and females in the proportion who said they buy clothing from their mobile device at the 0.05 level of​ significance.