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 300 homeowners in danger of foreclosure in the time period surrounding the enactment of this law. Of the 130 who fell behind just before the law was enacted, 25 were able to avoid foreclosure, and of 170 who fell behind just after the law was enacted, 42 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?

To calculate the p-value, we first need to calculate the proportions p1 and p2.

p1 = 25/130 = 0.1923 (proportion of delinquent borrowers who avoid foreclosure just before the law) p2 = 42/170 = 0.2471 (proportion of delinquent borrowers who avoid foreclosure just after the law)

We are testing the null hypothesis that the proportions are the same before and after the law, against the alternative hypothesis that the proportion after the law is greater.

We can use a two-proportion z-test to calculate the test statistic and the p-value. The formula for the test statistic is:

z = (p1 - p2) / sqrt [ p(1 - p) * (1/n1 + 1/n2) ]

where p is the pooled sample proportion, calculated as (x1 + x2) / (n1 + n2), where x1 and x2 are the number of successes in each sample, and n1 and n2 are the sizes of the two samples.

p = (25 + 42) / (130 + 170) = 67/300 = 0.2233

So, the test statistic z is:

z = (0.1923 - 0.2471) / sqrt [ 0.2233(1 - 0.2233) * (1/130 + 1/170) ] = -1.36

The p-value is the probability of getting a test statistic as extreme as -1.36 or more, assuming the null hypothesis is true. We look this up in a standard normal (Z) table or use a calculator, and find that the p-value is 0.0871.

Therefore, the appropriate p-value to verify the claim is 0.0871.

To calculate the p-value, we first need to calculate the proportions p1 and p2.

p1 = 25/130 = 0.1923 (proportion of delinquent borrowers who avoid foreclosure just before the law) p2 = 42/170 = 0.2471 (proportion of delinquent borrowers who avoid foreclosure just after the law)

We are testing the null hypothesis that the proportions are the same before and after the law, against the alternative hypothesis that the proportion after the law is greater.

We can use a two-proportion z-test to calculate the test statistic and the p-value. The formula for the test statistic is:

z = (p1 - p2) / sqrt [ p(1 - p) * (1/n1 + 1/n2) ]

where p is the pooled sample proportion, calculated as:

p = (x1 + x2) / (n1 + n2) = (25 + 42) / (130 + 170) = 0.2233

Substituting the values into the formula gives:

z = (0.1923 - 0.2471) / sqrt [ 0.2233(1 - 0.2233) * (1/130 + 1/170) ] = -1.36

The p-value is the probability of getting a test statistic as extreme as -1.36 or more so, assuming the null hypothesis is true. We look this up in a standard normal (Z) table or use a calculator to find that the p-value is 0.0871.

Therefore, the appropriate p-value to verify the claim is 0.0871.