Question

In the book Essentials of Marketing Research, William R. Dillon, Thomas J. Madden, and Neil H. Firtle discuss a research proposal in which a telephone company wants to determine whether the appeal of a new security system varies between homeowners and renters. Independent samples of 140 homeowners and 60 renters are randomly selected. Each respondent views a TV pilot in which a test ad for the new security system is embedded twice. Afterward, each respondent is interviewed to find out whether he or she would purchase the security system.

Results show that 25 out of the 140 homeowners definitely would buy the security system, while 9 out of the 60 renters definitely would buy the system.

a. Letting p₁ be the population proportion of homeowners who would buy the security system, and letting p₂ be the population proportion of renters who would buy the security system, set up the null and alternative hypotheses needed to determine whether the proportion of homeowners who would buy the security system differs from the proportion of renters who would buy the security system.

H0: p1−p2p1−p2(Click to select)><=≠≤≥ 0

Ha: p1−p2p1−p2(Click to select)=≠≤<>≥ 0

b. Find the test statistic z and the p-value for testing the hypotheses of part a. (α=0.05)

z= , p-value=

c. Determine the rejection rule:

    Critical Value Rule:

     Reject H₀ if (Click to select)t> t alpha/2 or t < -t alpha/2t > t alphat < t alpha/2z> z alpha/2 or z < -z alpha/2z > z alphaz < z alpha

d. What is the meaning of p-value for the hypothesis test in part a if pˆ1−pˆ2p^1−p^2 = 0.1786-0.15= 0.0286? (Hint: p-value is the probability that we will get the sample values (or more extreme) from a population where the null hypothesis holds. THIS IS A TWO-SIDED TEST.)

(Click to select)The probability that the difference in sample proportions is greater than 0.0286.The probability that the difference in sample proportions is less than 0.0286The probability that the difference in sample proportions is more extreme than 0.0286 if the difference in population proportions is equal to 0.

e. We have (Click to select)someextremely strongvery strongnostrong evidence that the proportions of homeowners and renters differ.

f. Calculate a 90 percent confidence interval for the difference between the proportions of homeowners and renters who would buy the security system.

≤ p1−p2p1−p2 ≤

g. Can we conclude at the 90% confidence that the difference between the proportions of homeowners and renters who would buy the security system is greater than 0?

(Click to select)Yes, because the whole interval is above 0. No, because 0 is within the confidence interval. Yes, because the whole interval is below 0.

h. Can we conclude at the 90% confidence that the difference between the proportions of homeowners and renters who would buy the security system is less than 0.1?

(Click to select)Yes, because the whole interval is above 0.1. Yes, because the whole interval is below 0.1.No, because 0.1 is within the confidence interval.

Answer 1

For sample 1, we have that the sample size is N1=140, the number of favorable cases is X1=25, so then the sample proportion is p^1=N1/X1 =25/140=0.1786
For sample 2, we have that the sample size is N2 =60, the number of favorable cases is X2 =9, so then the sample proportion is p^2 = X2/N2 = 9/60= 0.15

The value of the pooled proportion is computed as p- = X1+X2/N1+N2 = (25+9)/(140+60)= 34/200= 0.17
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
​Ha:p1p2
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

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(4) Decision about the null hypothesis

Since it is observed that ∣z∣=0.493≤zc=1.96, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.6221, and since p=0.6221≥0.05, it is concluded that the null hypothesis is not rejected.

The probability that the difference in sample proportions is more extreme than 0.0286 if the difference in population proportions is equal to 0.

No strong evidence that the proportions of homeowners and renters differ.

The 90% confidence interval for p1−p2 is: −0.064<p1−p2 <0.121.

No, because 0 is within the confidence interval.

No, because 0.1 is within the confidence interval.