In the book Essentials of Marketing Research, WilliamR. Dillon, Thomas J. Madden, and Neil H. Firtle discuss a researchproposal in which a telephone company wants to determine whetherthe appeal of a new security system varies between homeowners andrenters. Independent samples of 140 homeowners and 60 renters arerandomly selected. Each respondent views a TV pilot in which a testad for the new security system is embedded twice. Afterward, eachrespondent is interviewed to find out whether he or she wouldpurchase the security system.
Results show that 25 out of the 140 homeowners definitely wouldbuy the security system, while 9 out of the 60 renters definitelywould buy the system.
a. Letting p1 be the population proportionof homeowners who would buy the security system, and lettingp2 be the population proportion of renters whowould buy the security system, set up the null and alternativehypotheses needed to determine whether the proportion of homeownerswho would buy the security system differs from the proportion ofrenters 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-valuefor testing the hypotheses of part a. (?=0.05)
z= , p-value=
c. Determine the rejection rule:
Critical ValueRule:
Reject H0 if (Click toselect)t> t alpha/2 or t < -t alpha/2t > t alphat < talpha/2z> z alpha/2 or z < -z alpha/2z > z alphaz < zalpha
d. What is the meaning of p-value for the hypothesis test inpart a if pˆ1?pˆ2p^1?p^2 = 0.1786-0.15= 0.0286? (Hint: p-value isthe probability that we will get the sample values (or moreextreme) from a population where the null hypothesis holds. THIS ISA TWO-SIDED TEST.)
(Click to select)The probability that the difference in sampleproportions is greater than 0.0286.The probability that thedifference in sample proportions is less than 0.0286The probabilitythat the difference in sample proportions is more extreme than0.0286 if the difference in population proportions is equal to0.
e. We have (Click to select)someextremely strongverystrongnostrong evidence that the proportions of homeowners andrenters differ.
f. Calculate a 90 percent confidence interval for the differencebetween the proportions of homeowners and renters who would buy thesecurity system.
? p1?p2p1?p2 ?
g. Can we conclude at the 90% confidence that the differencebetween the proportions of homeowners and renters who would buy thesecurity 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 wholeinterval is below 0.
h. Can we conclude at the 90% confidence that the differencebetween the proportions of homeowners and renters who would buy thesecurity 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 iswithin the confidence interval.
