Question

Answer the following questions:

1. Loans: Before lending someone money, banks must decide whether they believe the applicant will repay the loan. One strategy used is a point system. Loan officers assess information about the applicant, totaling points they award for the person’s income level, credit history, current debt burden, and so on. The higher the point total, the more convinced the bank is that it’s safe to make the loan. Ant applicant with a lower point total than certain a cutoff score is denied.
We can think this decision as a hypothesis test. Since the bank makes its profit from the interest collected on repaid loans, their null hypothesis is that the applicant will repay the loan and therefore should get the money. Only if the person’s score falls below the minimum cutoff will the bank reject the null and deny the loan. This system is reasonably reliable, but, of course sometimes there are mistakes.
                                                                                                               
a. When the person defaults on a loan, which type of error did the bank make?
b. Which kind of error is it when the bank misses an opportunity to make a loan to someone who would have repaid it?
c. Suppose the bank decides to lower the cut off score from 250 points to 200 points. Is that analogous to choosing a higher or lower of α for a hypothesis test?
d. What impact does this change in the cutoff value have on the chance of each type of error?
                  
2. A new reading program may reduce the number of elementary school students who read below grade level. The company that developed this program supplied materials and teacher training for a large-scale involving 8500 children in several different school districts. Statistical analysis of the results showed that the percentage of students who did not attain the grade-level standard was reduced from 15.9% to 15.1%. The hypothesis that the new reading program produced no improvement was rejected with P-value of 0.023.

Explain what P-value means in this context.
Even though this reading method has been shown to be significantly better, why might you not recommend that your local school adopt it?

3. Anyone who plays or watches has heard of the “home field advantage”. Teams tend to win more often when they play at home. Or do they?
If there were no home field advantage, the home teams would win about half of all games played. In the 2006 major League Baseball season, there were 2419 regular-season games. (One rained-out game was never made up.) It turns out that the home team won 1327 out of the 2419 games, or 54.86% of the time.
                                                                                                                  
Question: Could this deviation from 50% be explained just from natural sampling variability, or is it evidence to suggest that there really is a home field advantage, at least in professional baseball?

a. State H0 and Ha.
b. Think about the assumptions and check the appropriate conditions
c. Find z score
d. Find P value
e. Explain what P-value means in this context.
f. Is your test statistically significant? What do you think of practical significance?
g. State your conclusion.(It is not enough only to state reject or not reject null hypothesis)(State whether there is a home field advantage or not)
h. Find 95% confidence interval
i. State your conclusion. (State the interval % of home teams winning and state whether there is a home field advantage or not)
j. Are your conclusion from f and h match?

4. Used cars 2007: Classified ads in the Ithica Journal offered several used Toyota Corollas for sale. Please check Excel spread sheet for data.
                                                                     
a. Make a scatterplot for the data
b. Describe the association between Age and Price of a used Corolla
c. Do you think a linear model is appropriate? Why or why not? Explain
d. Find r2. Find r.
e. Explain the meaning of r2 and r in this context.
f. Why does not this model explain 100% of the variability in the price of a used Corolla?
g. Find the equation of the regression line
h. Explain the meaning of the slope of the line
i. Explain the meaning of the y-intercept of the line
j. If you want to see a 7-year-old Corolla, what price seems appropriate?
k. You have a chance to buy one of the two cars. They are about the same age and appear to be in equally good condition. Would you rather buy the one with a positive residual or the one with a negative residual? Explain (Residual = actual price – predicted price)
l. You see a “For Sale” sign on a 10-year-old Corolla stating the asking price as $3500. What is the residual? Explain what it means in this context
m. Would this regression model be useful in establishing a fair price for a 20-year-old car? Explain.

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1.

a. When the person defaults on a loan, which type of error did the bank make?
Type 1 Error i.e. rejecting the null hypothesis when it is actually true.

b. Which kind of error is it when the bank misses an opportunity to make a loan to someone who would have repaid it?
Type II Error. Failing to reject the null hypotheses when it is actually False.

c. Suppose the bank decides to lower the cut off score from 250 points to 200 points. Is that analogous to choosing a higher or lower of α for a hypothesis test?
No. Reducing the cut off score is not similar to choosing a higher or lower alpha. Alpha indicates a confidence in the test and is independent of the test statistic. Apparently it is the chance that the bank is willing to take in order to commit a type I error.

d. What impact does this change in the cutoff value have on the chance of each type of error?
The change will not have any effect on the type I error neither does it have any effect on the type II error....

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