regression analysis problem

220 REGRESSION ANALYSIS AND FORECASTING

TABLE E3.1 Days that Ozone Levels Exceed 20 ppm and Seasonal Meteorological Index

Year Days Index

1976 91 16.7 1977 105 17.1 1978 106 18.2 1979 108 18.1 1980 88 17.2 1981 91 18.2 1982 58 16.0 1983 82 17.2 1984 81 18.0 1985 65 17.2 1986 61 16.9 1987 48 17.1 1988 61 18.2 1989 43 17.3 1990 33 17.5 1991 36 16.6

3.2 Montgomery, Peck, and Vining (2012) present data on the number of pounds of steam used per month at a plant. Steam usage is thought to be related to the average monthly ambient temperature. The past year’s usages and temperatures are shown in Table E3.2.

TABLE E3.2 Monthly Steam Usage and Average Ambient Temperature

Month Temperature

(◦F) Usage/1000 Month Temperature

(◦F) Usage/1000

January 21 185.79 July 68 621.55 February 24 214.47 August 74 675.06 March 32 288.03 September 62 562.03 April 47 424.84 October 50 452.93 May 50 454.68 November 41 369.95 June 59 539.03 December 30 273.98

a. Fit a simple linear regression model to the data. b. Test for significance of regression. c. Analyze the residuals from this model.

EXERCISES 221

d. Plant management believes that an increase in average ambient temperature of one degree will increase average monthly steam consumption by 10,000 lb. Do the data support this statement?

e. Construct a 99% prediction interval on steam usage in a month with average ambient temperature of 58◦F.

3.3 On March 1, 1984, the Wall Street Journal published a survey of television advertisements conducted by Video Board Tests, Inc., a New York ad-testing company that interviewed 4000 adults. These people were regular product users who were asked to cite a commer- cial they had seen for that product category in the past week. In this case, the response is the number of millions of retained impressions per week. The predictor variable is the amount of money spent by the firm on advertising. The data are in Table E3.3.

TABLE E3.3 Number of Retained Impressions and Advertising Expenditures

Amount Spent Retained Impressions Firm (Millions) per Week (Millions)

Miller Lite 50.1 32.1 Pepsi 74.1 99.6 Stroh’s 19.3 11.7 Federal Express 22.9 21.9 Burger King 82.4 60.8 Coca-Cola 40.1 78.6 McDonald’s 185.9 92.4 MCI 26.9 50.7 Diet Cola 20.4 21.4 Ford 166.2 40.1 Levi’s 27 40.8 Bud Lite 45.6 10.4 ATT Bell 154.9 88.9 Calvin Klein 5 12 Wendy’s 49.7 29.2 Polaroid 26.9 38 Shasta 5.7 10 Meow Mix 7.6 12.3 Oscar Meyer 9.2 23.4 Crest 32.4 71.1 Kibbles N Bits 6.1 4.4

222 REGRESSION ANALYSIS AND FORECASTING

a. Fit the simple linear regression model to these data. b. Is there a significant relationship between the amount that a com-

pany spends on advertising and retained impressions? Justify your answer statistically.

c. Analyze the residuals from this model. d. Construct the 95% confidence intervals on the regression coeffi-

cients. e. Give the 95% confidence and prediction intervals for the number

of retained impressions for MCI.

3.4 Suppose that we have fit the straight-line regression model ŷ = "̂0 + "̂1x1, but the response is affected by a second variable x2 such that the true regression function is

E(y) = "0 + "1x1 + "2x2

a. Is the least squares estimator of the slope in the original simple linear regression model unbiased?

b. Show the bias in "̂1.

3.5 Suppose that we are fitting a straight line and wish to make the standard error of the slope as small as possible. Suppose that the “region of interest” for x is −1 ≤ x ≤ 1. Where should the obser- vations x1, x2,… , xn be taken? Discuss the practical aspects of this data collection plan.

3.6 Consider the simple linear regression model

y = "0 + "1x + #,

where the intercept "0 is known. a. Find the least squares estimator of "1 for this model. Does this

answer seem reasonable? b. What is the variance of the slope ("̂1) for the least squares esti-

mator found in part a? c. Find a 100(1 − $) percent CI for "1. Is this interval narrower

than the estimator for the case where both slope and intercept are unknown?

3.7 The quality of Pinot Noir wine is thought to be related to the proper- ties of clarity, aroma, body, flavor, and oakiness. Data for 38 wines are given in Table E3.4.