Question

A market research survey asked the question “Do you enjoy shopping for clothing for yourself?” to 542 male adults and 238 of them said they did. When 543 female adults were asked the same question, 276 of them indicated that they enjoyed shopping for clothing for themselves. (Source: USA Today)

1. Is there a significant difference between the proportion of males and females who enjoy shopping for clothing for themselves at the 0.05 level of significance? Perform a hypothesis test with all its steps to make an informed decision. Show your work or calculator commands.

2. Find a 95% and confidence interval for the difference between the proportions of males and females who enjoy shopping for clothing for themselves. Show your work or calculator commands. Does this interval support your decision in part a?

Answer 1

For sample 1,

we have that the sample size is N1​=542,

the number of favorable cases is X1​=238,

so then the sample proportion is p1=​​X1/N1 ​​= 238/542 ​= 0.4391

For sample 2, we have that the sample size is N2​=543,

the number of favorable cases is X2​=276,

so then the sample proportion is p2 ​= ​X2/N2​​ = 276/543​=0.5083

The value of the pooled proportion is computed as p ​=(​X1​+X2)/(N1​+N2)​​=(238+276)/(542+543)​=0.4737

Also, the given significance level is α=0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:p1​=p2​ i.e. There is not a significant difference between the proportion of males and females who enjoy shopping for clothing for themselves

Ha:p1​≠ p2 i.e. There is a significant difference between the proportion of males and females who enjoy shopping for clothing for themselves

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=0.05, and the critical value for a two-tailed test is zc​=1.96.

The rejection region for this two-tailed test is R={z:∣z∣>1.96}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that ∣z∣=2.282>zc​=1.96, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0225, and since p=0.0225<0.05, it is concluded that the null hypothesis is rejected.

Here p value is computed using normal distribution with mean 0 and variance 1

p = P[z>2.282] + P[z<-2.282]

p = 2*P[z>2.282]

(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 0.05 significance level.

Hence There is a significant difference between the proportion of males and females who enjoy shopping for clothing for themselves

Confidence Interval

The 95% confidence interval p1​−p2​

The 95% confidence interval for p1​−p2​ is

−0.128<p1​−p2​<−0.01.