SECTION 10.2 Exercises

For Exercises 10.39 and 10.40, see page 514.

Many of the following exercises require use of software that will calculate the intervals required for predicting mean response and individual response.

Question 10.41

10.41 More on public university tuition.

Refer to Exercise 10.15 (pages 504–505).

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  1. The tuition at BusStat U was $8800 in 2008. Find the 95% prediction interval for its tuition in 2013.
  2. The tuition at Moneypit U was $15,700 in 2008. Find the 95% prediction interval for its tuition in 2013.
  3. Compare the widths of these two intervals. Which is wider and why?

10.41

(a) (6899, 13385). (b) (13024, 20400). (c) Moneypit U is wider because it is farther from the mean for Y2008.

Question 10.42

10.42 More on assessment value versus sales price.

Refer to Exercises 10.13 and 10.14 (pages 503–504). Suppose we’re interested in determining whether the population regression line differs from . We’ll look at this three ways.

hsales

  1. Construct a 95% confidence interval for each property in the data set. If the model is reasonable, then the assessed value used to predict the sales price should be in the interval. Is this true for all cases?
  2. The model means and . Test these two hypotheses. Is there enough evidence to reject either of these two hypotheses?
  3. Recall that not rejecting does not imply is true. A test of “equivalence” would be a more appropriate method to assess similarity. Suppose that, for the slope, a difference within is considered not different. Construct a 90% confidence interval for the slope and see if it falls entirely within the interval . If it does, we would conclude that the slope is not different from 1. What is your conclusion using this method?

Question 10.43

10.43 Predicting 2013 tuition from 2008 tuition.

Refer to Exercise 10.15 (pages 504–505).

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  1. Find a 95% confidence interval for the mean tuition amount corresponding to a 2008 tuition of $7750.
  2. Find a 95% prediction interval for a future response corresponding to a 2008 tuition of $7750.
  3. Write a short paragraph interpreting the meaning of the intervals in terms of public universities.
  4. Do you think that these results can be applied to private universities? Explain why or why not.

10.43

(a) (8574, 9709). (b) (5900, 12384). (c) For all public universities with tuition of $7750 in 2008, the mean tuition in 2013 will be between $8574 and $9709 with 95% confidence. For an individual public university with tuition of $7750 in 2008, the prediction tuition in 2013 will be between $5900 and $12,384 with 95% confidence. (d) No, private universities likely are different than public universities.

Question 10.44

10.44 Predicting 2013 tuition from 2000 tuition.

Refer to Exercise 10.21 (page 506).

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  1. Find a 95% confidence interval for the mean tuition amount corresponding to a 2000 tuition of $3872.
  2. Find a 95% prediction interval for a future response corresponding to a 2000 tuition of $3872.
  3. Write a short paragraph interpreting the meaning of the intervals in terms of public universities.
  4. Do you think that these results can be applied to private universities? Explain why or why not.

Question 10.45

10.45 Compare the estimates.

Case 18 in Table 10.3 (Purdue) has a 2000 tuition of $3872 and a 2008 tuition of $7750. A predicted 2013 tuition amount based on 2008 tuition was computed in Exercise 10.43, while one based on the 2000 tuition was computed in Exercise 10.44. Compare these two estimates and explain why they differ. Use the idea of a prediction interval to interpret these results.

10.45

Using the 2008 tuition the prediction interval is (5900, 12384), using the 2000 tuition the prediction interval is (5173, 12980). The intervals are different because they are based on two different models using different predictors. That said, the intervals don’t differ that much, with the 1st interval entirely in the 2nd interval. It could be that the 2008 data better predict the 2013 tuition because such data are closer in time to 2013 and thus give a better estimate than the 2000 data.

Question 10.46

10.46 Is the price right?

Refer to Exercise 10.31 (page 508), where the relationship between the size of a home and its selling price is examined.

hsize

  1. Suppose that you have a client who is thinking about purchasing a home in this area that is 1500 square feet in size. The asking price is $140,000. What advice would you give this client?
  2. Answer the same question for a client who is looking at a 1200-square-foot home that is selling for $100,000.

517

Question 10.47

10.47 Predicting income from age.

Figures 10.11 and 10.12 (pages 506 and 507) analyze data on the age and income of 5712 men between the ages of 25 and 65. Here is Minitab output predicting the income for ages 30, 40, 50, and 60 years:

Predicted Values

Fit SE Fit 95% CI 95% PI
51638 948 (49780, 53496) (−41735, 145010)
60559 637 (59311, 61807) (−32803, 153921)
69480 822 (67870, 71091) (−23888, 162848)
78401 1307 (75840, 80963) (−14988, 171790)
  1. Use the regression line from Figure 10.11 (page 506) to verify the “Fit” for age 30 years.
  2. Give a 95% confidence interval for the income of all 30-year-old men.
  3. Joseph is 30 years old. You don’t know his income, so give a 95% prediction interval based on his age alone. How useful do you think this interval is?

10.47

(a) . (b) (49780, 53496). (c) (−41735, 145010). The interval isn’t very useful; we could have guessed he was in a similar range without any statistics.

Question 10.48

10.48 Predict what?

The two 95% intervals for the income of 30-year-olds given in Exercise 10.47 are very different. Explain briefly to someone who knows no statistics why the second interval is so much wider than the first. Start by looking at 30-year-olds in Figure 10.11 (page 506).

Question 10.49

10.49 Predicting income from age, continued.

Use the computer outputs in Figure 10.12 (page 507) and Exercise 10.47 to give a 99% confidence interval for the mean income of all 40-year-old men.

10.49

(58,917, 62,201).

Question 10.50

10.50 T-bills and inflation.

Figure 10.16 (page 514) gives part of a regression analysis of the data in Table 10.1 relating the return on Treasury bills to the rate of inflation. The output includes prediction of the T-bill return when the inflation rate is 2.25%.

  1. Use the output to give a 90% confidence interval for the mean return on T-bills in all years having 2.25% inflation.
  2. You think that next year’s inflation rate will be 2.25%. It isn’t possible, without complicated arithmetic, to give a 90% prediction interval for next year’s T-bill return based on the output displayed. Why not?

Question 10.51

10.51 Two confidence intervals.

The data used for Exercise 10.47 include 195 men 30 years old. The mean income of these men is and the standard deviation of these 195 incomes is .

  1. Use the one-sample procedure to give a 95% confidence interval for the mean income of 30-year-old men.
  2. Why is this interval different from the 95% confidence interval for in the regression output?

(Hint: What data are used by each method?)

10.51

(a) (44,446, 55,314). (b) The interval from part (a) only includes information from the 195 30-year-old men, while the interval for the mean response in the output uses the data from all 5712 men to give a better estimate for the 30-year-olds.

Question 10.52

10.52 Size and selling price of houses.

Table 10.5 (page 509) gives data on the size in square feet of a random sample of houses sold in a Midwest city along with their selling prices.

hsize

  1. Find the mean size of these houses and also their mean selling price . Give the equation of the least-squares regression line for predicting price from size, and use it to predict the selling price of a house of mean size. (You knew the answer, right?)
  2. Jasmine and Woodie are selling a house whose size is equal to the mean of our sample. Give an interval that predicts the price they will receive with 95% confidence.

Question 10.53

10.53 Beer and blood alcohol.

Exercise 10.34 (page 509) gives data from measuring the blood alcohol content (BAC) of students 30 minutes after they drank an assigned number of cans of beer. Steve thinks he can drive legally 30 minutes after he drinks five beers. The legal limit is . Give a 90% prediction interval for Steve’s BAC. Can he be confident he won’t be arrested if he drives and is stopped?

bac

10.53

(a) (0.04, 0.1142). Because the prediction interval includes values over 0.08, he can’t be confident he won’t be arrested.

Question 10.54

10.54 Selling a large house.

Among the houses for which we have data in Table 10.5 (page 509), just four have floor areas of 1800 square feet or more. Give a 90% confidence interval for the mean selling price of houses with floor areas of 1800 square feet or more.

hsize