Question

A survey of 1000 adults from a certain region​ asked, "Do you enjoy shopping for clothing for​ yourself?" The results indicated that 59​% of the females enjoyed shopping for clothing for themselves as compared to 51​% of the males. The sample sizes of males and females were not provided. Suppose that of 600 ​females, 354 said that they enjoyed shopping for clothing for themselves while of 400 ​males, 204 said that they enjoyed shopping for clothing for themselves. Complete parts​ (a) through​ (d) below.

a. Is there evidence of a difference between males and females in the proportion who enjoy shopping for clothing for themselves at the 0.1 level of​ significance? State the null and alternative​ hypotheses, where pi 1 is the population proportion of females that enjoy shopping for themselves and pi 2 is the population proportion of males that enjoy shopping for themselves.

Determine the value of the test statistic.

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

The critical​ value(s) is​ (are) nothing

find the p value

Construct and interpret a 90​% confidence interval estimate for the difference between the proportion of males and females who enjoy shopping for clothing for themselves.

What are the answers to​ (a) through​ (c) if 212 males enjoyed shopping for clothing for​ themselves?

Is there evidence of a difference between males and females in the proportion who enjoy shopping for clothing for themselves at the 0.1 level of​ significance? State the null and alternative​ hypotheses, where pi 1π1 is the population proportion of females that enjoy shopping for themselves and pi 2π2 is the population proportion of males that enjoy shopping for themselves.

Determine the value of the test statistic.

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

The critical​ value(s) is​ (are)

Find the​ p-value and interpret its meaning.

Construct and interpret a 90​% confidence interval estimate for the difference between the proportion of males and females who enjoy shopping for clothing for themselves.

Answer 1

For Female, we have that the sample size is N1​=600, the number of favorable cases is X1​=354, so then the sample proportion is p^​1​=X1/​N1​=354/600 ​=0.59

For sample 2, we have that the sample size is N2​=400, the number of favorable cases is X2​=204, so then the sample proportion is p^​2​=​X2/N2​​=204/400​=0.51

The value of the pooled proportion is computed as \p¯​=X1​+X2/( N1​+N2​) ​​= 354+204​/(600+400)=0.558

Also, the given significance level is α=0.1.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:p1​=p2​

Ha:p1​≠p2​

This corresponds to a two-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

Based on the information provided, the significance level is α=0.1, and the critical value for a two-tailed test is CRITICAL VALUE zc​=1.64.

The rejection region for this two-tailed test is R={z:∣z∣>1.64}

(3) Test Statistics

The z-statistic is computed as follows:

p^​1​−p^​2​​=0.59−0.51​sqrt(0.558⋅(1−0.558)(1/600+1/400)) ​=2.496

(4) Decision about the null hypothesis

Since it is observed that ∣z∣=2.496>zc​=1.64, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0126, and since p=0.0126<0.1, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion p1​ is different than p2​, at the 0.1 significance level.

Confidence Interval

The 90% confidence interval for is: 0.027 < p_1- p_2 < 0.133

NOTE: I HAVE DONE THE FIRST FOUR QUESTIONS. PLEASE REPOST THE SECOND PART ALONG WITH THE QUESTION. THANK YOU.