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.

Letting p1 be the proportion of homeowners who would buy the security system, and letting p2 be the 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.

Answer 1

Answer:

Null hypothesis:$p_{1} = p_{2}$  

Alternative hypothesis:$p_{1} \neq p_{2}$  

$z=\frac{0.179-0.15}{\sqrt{0.17(1-0.17)(\frac{1}{140}+\frac{1}{60})}}=0.500$  

$p_v =2*P(Z>0.500)=0.617$  

So the p value is a very low value and using any significance level for example $\alpha=0.05, 0,1,0.15$ always $p_v>\alpha$ so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can say the two proportions NOT differs significantly.  

Step-by-step explanation:

Data given and notation  

$X_{1}=25$ represent the number of homeowners who would buy the security system

$X_{2}=9$ represent the number of renters who would buy the security system

$n_{1}=140$ sample 1

$n_{2}=60$ sample 2

$p_{1}=\frac{25}{140}=0.179$ represent the proportion of homeowners who would buy the security system

$p_{2}=\frac{9}{60}= 0.15$ represent the proportion of renters who would buy the security system

z would represent the statistic (variable of interest)  

$p_v$ represent the value for the test (variable of interest)  

Concepts and formulas to use  

We need to conduct a hypothesis in order to check if the two proportions differs , the system of hypothesis would be:  

Null hypothesis:$p_{1} = p_{2}$  

Alternative hypothesis:$p_{1} \neq p_{2}$  

We need to apply a z test to compare proportions, and the statistic is given by:  

$z=\frac{p_{1}-p_{2}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}$   (1)  

Where $\hat p=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{25+9}{140+60}=0.17$  

Calculate the statistic  

Replacing in formula (1) the values obtained we got this:  

$z=\frac{0.179-0.15}{\sqrt{0.17(1-0.17)(\frac{1}{140}+\frac{1}{60})}}=0.500$  

Statistical decision

For this case we don't have a significance level provided $\alpha$, but we can calculate the p value for this test.    

Since is a two sided test the p value would be:  

$p_v =2*P(Z>0.500)=0.617$  

So the p value is a very low value and using any significance level for example $\alpha=0.05, 0,1,0.15$ always $p_v>\alpha$ so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can say the two proportions NOT differs significantly.