Question

While her husband spent 2½ hours picking out new speakers, a statistician decided to determine whether the percent of men who enjoy shopping for electronic equipment is higher than the percent of women who enjoy shopping for electronic equipment. The population was Saturday afternoon shoppers. Out of 65 men, 25 said they enjoyed the activity. Eight of the 22 women surveyed claimed to enjoy the activity. Interpret the results of the survey. Conduct a hypothesis test at the 5% level. Let the subscript m = men and w = women.
NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

State the distribution to use for the test. (Round your answers to four decimal places.)

P'_m − P'_w ~N : 0, ________ ?

Answer 1

Solution:

We shall use the standard normal distribution (z-test) to use for the test.

The null and alternative hypotheses are as follows:

i.e. Population proportions for men and women who enjoy shopping for electronic equipments are same.

i.e. The population proportion of men who enjoy shopping for electronic equipments is greater than that of women.

To test the hypothesis we shall use two sample z-test for equality of two proportions. The test statistic is given as follows:

Where, p̂_{​​​​​m} and p̂_{​​​​​f} are sample proportions for males and females respectively. n_{​​​​​​1} and n_{​​​​​​2} are sample sizes.

Q = 1 - P

Sample proportion of men who enjoy shopping for electronic equipments is,

Sample proportion of women who enjoy shopping for electronic equipments is,

n_{​​​​​​1} = 65 and n_{​​​​​​2} = 22

Q = 1 - 0.3793 = 0.6207

The value of the test statistic is 0.1753.

Since, our test is right-tailed test, therefore we shall obtain right-tailed p-value for the test statistic, which is given as follows:

p-value = P(Z > value of the test statistic)

p-value = P(Z > 0.1753)

p-value = 0.4304

The p-value is 0.4304.

We make decision rule as follows:

If p-value is greater than the significance level, then we fail to reject the null hypothesis (H_{​​​​​​0}) at given significance level.

If p-value is less than the significance level, then we reject the null hypothesis (H_{​​​​​​0}) at given significance level.

We have, p-value = 0.4304 and significance level = 5% = 0.05

(0.4304 > 0.05)

Since, p-value is greater than the significance level of 0.05, therefore we shall be fail to reject the null hypothesis (H_{​​​​​​0}) at 0.05 significance level.

Conclusion : At significance level of 5%, there is not sufficient evidence to conclude that the percent of men who enjoy shopping for electronic equipment is higher than the percent of women who enjoy shopping for electronic equipment.