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In August 2010, Massachusetts enacted a 150-day right-to-cure period that mandates that lenders give homeowners who fall behind on their mortgage an extra five months to become current before beginning foreclosure proceedings. Policymakers claimed that the policy would result in a higher proportion of delinquent borrowers becoming current on their mortgages. To test this claim, researchers took a sample of 244 homeowners in danger of foreclosure in the time period surrounding the enactment of this law. Of the 100 who fell behind just before the law was enacted, 30 were able to avoid foreclosure, and of 144 who fell behind just after the law was enacted, 48 were able to avoid foreclosure. Let p1 and p2 represent the proportion of delinquent borrowers who avoid foreclosure just before and just after the right-to-cure law is enacted, respectively. Which of the following is the appropriate p-value to verify the claim?
a. 0.2915
b. 0.5570
c. 0.5832
d. 0.7084

Respuesta :

Answer:

[tex]z=\frac{0.333-0.3}{\sqrt{0.320(1-0.320)(\frac{1}{100}+\frac{1}{144})}}=0.544[/tex]    

The p value for this case would be given by:

[tex]p_v =P(Z>0.544)\approx 0.2915[/tex]    

The best option would be:

a. 0.2915

Step-by-step explanation:

Information given

[tex]X_{1}=30[/tex] represent the number of people that were able to avoid foreclosure from who fell behind just before the law was enacted

[tex]X_{2}=48[/tex] represent the number of people  that were able to avoid foreclosure from who fell behind  just after the law was enacted

[tex]n_{1}=100[/tex] sample 1 selected  

[tex]n_{2}=144[/tex] sample 2 selected  

[tex]p_{1}=\frac{30}{100}=0.3[/tex] represent the proportion estimated of delinquent borrowers who avoid foreclosure just before

[tex]p_{2}=\frac{48}{144}=0.333[/tex] represent the proportion estimated of delinquent borrowers who avoid foreclosure just after

[tex]\hat p[/tex] represent the pooled estimate of p

z would represent the statistic

[tex]p_v[/tex] represent the value for the test

System of hypothesis

Policymakers claimed that the policy would result in a higher proportion of delinquent borrowers becoming current on their mortgages , so then the system of hypothesis are:    

Null hypothesis:[tex]p_{2} \leq p_{1}[/tex]    

Alternative hypothesis:[tex]p_{2} > p_{1}[/tex]    

The statistic is given by:

[tex]z=\frac{p_{1}-p_{2}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}[/tex]   (1)  

Where [tex]\hat p=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{30+48}{100+144}=0.320[/tex]  

Replacing we got:

[tex]z=\frac{0.333-0.3}{\sqrt{0.320(1-0.320)(\frac{1}{100}+\frac{1}{144})}}=0.544[/tex]    

The p value for this case would be given by:

[tex]p_v =P(Z>0.544)\approx 0.2915[/tex]    

The best option would be:

a. 0.2915

The appropriate p-value to verify the claim is 0.2915 and this can be determined by using the formula of z-score.

Given :

  • The sample size is 244.
  • 100 who fell behind just before the law was enacted, 30 were able to avoid foreclosure, and of 144 who fell behind just after the law was enacted, 48 were able to avoid foreclosure.

The formula of z-score can be used in order to determine the appropriate p-value to verify the claim.

[tex]z = \dfrac{p_1-p_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1}+\frac{1}{n_2})} }[/tex]     --- (1)

Now, the value of [tex]\hat{p}[/tex] is calculated as:

[tex]\hat{p}=\dfrac{X_1+X_2}{n_1+n_2}[/tex]

[tex]\hat{p} = \dfrac{30+48}{100+144}=0.320[/tex]

Now, substitute the values of all the known terms in the expression (1).

[tex]z = \dfrac{0.333-0.3}{\sqrt{0.320(1-0.320)(\frac{1}{100}+\frac{1}{144})} }[/tex]

Simplify the above expression.

[tex]z = 0.544[/tex]

Now, using the z-table the p-value can be calculated.

[tex]p_v=P(Z0.544)[/tex]

[tex]p_v\approx 0.2915[/tex]

Therefore, the correct option is a).

For more information, refer to the link given below:

https://brainly.com/question/13299273

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