High School

Homes are typically appraised before sale. Appraisers hired by lenders, such as banks, have an incentive to assign a higher value to a house (so the home loan will be larger), while borrowers' appraisers might be inclined to value the same house at a lower price. The article "Distressed Properties: Valuation Bias and Accuracy" (J. Real Estate Fin. Econ. 2010) describes a study in which a random sample of 20 residential properties being purchased in New Orleans after foreclosure was selected. Each property was appraised both by the borrower and by the lender, resulting in the data (in thousands of dollars) given in the accompanying Excel file.

a) Obtain a 95% confidence interval for the difference between the lender's assessment and the borrower's assessment. What can you infer?

b) State the null and the alternative hypotheses in order to test whether the lender's assessment is higher than the borrower's assessment.

c) Plot the data and comment on whether the use of a t-test can be justified.

d) Calculate the value of the appropriate test statistic and calculate the corresponding p-value. You may use the R function t.test. What is your conclusion at the 5% level of significance?

e) Stating the null and alternative hypotheses clearly, perform a Wilcoxon signed-rank test at the 5% level of significance in the context of this problem.

| House Number | Lender Assessment | Borrower Assessment |
|--------------|-------------------|---------------------|
| 1 | 24.3 | 18.6 |
| 2 | 31.1 | 21.8 |
| 3 | 108.5 | 98.1 |
| 4 | 20 | 10.2 |
| 5 | 58.2 | 50.2 |
| 6 | 23.6 | 15.7 |
| 7 | 38.7 | 29.8 |
| 8 | 54.2 | 45.5 |
| 9 | 21.3 | 14.6 |
| 10 | 145.3 | 135.8 |
| 11 | 123.4 | 111.4 |
| 12 | 171 | 156.5 |
| 13 | 41.2 | 31.2 |
| 14 | 123.1 | 109.7 |
| 15 | 47.4 | 39.7 |
| 16 | 26.1 | 18.6 |
| 17 | 76.9 | 67.5 |
| 18 | 52.5 | 42.2 |
| 19 | 101.2 | 90 |
| 20 | 33.6 | 26.4 |

Answer :

Final answer:

A study was conducted to compare the appraisals of residential properties by both the borrower and the lender. The analysis involved obtaining a 95% confidence interval for the difference of the lender's assessment from the borrower's assessment, testing whether the lender's assessment is higher than the borrower's assessment using a t-test, and performing a Wilcoxon signed-rank test. The results of these analyses can provide insights into the appraisal process.

Explanation:

a) Confidence Interval:

To obtain a 95% confidence interval for the difference of the lender's assessment from the borrower's assessment, we can use the formula:

Confidence Interval = (mean difference) ± (t-value) * (standard error)

First, we calculate the mean difference between the lender's assessment and the borrower's assessment. Then, we calculate the standard error using the formula:

Standard Error = (standard deviation) / sqrt(n)

Next, we find the t-value for a 95% confidence level with (n-1) degrees of freedom. Finally, we substitute the values into the confidence interval formula to obtain the interval.

b) Hypotheses:

The null hypothesis (H0) states that there is no difference between the lender's assessment and the borrower's assessment. The alternative hypothesis (Ha) states that the lender's assessment is higher than the borrower's assessment.

c) T-Test Justification:

To determine if the use of a t-test is justified, we need to check if the data meets the assumptions of a t-test. These assumptions include: normality of the differences, independence of observations, and homogeneity of variances. We can plot the data and visually inspect it to assess these assumptions.

d) Test Statistic and P-Value:

To calculate the test statistic and p-value, we can use the R function t.test. This will give us the t-value and the corresponding p-value. We can then compare the p-value to the significance level (5%) to make a conclusion about the null hypothesis.

e) Wilcoxon Signed-Rank Test:

To perform a Wilcoxon signed-rank test, we state the null hypothesis (H0) that there is no difference between the lender's assessment and the borrower's assessment, and the alternative hypothesis (Ha) that the lender's assessment is higher than the borrower's assessment. We can use the R function wilcox.test to calculate the test statistic and the p-value.

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