the data support this statement?d. Construct a 99% prediction interval on steam usage in a month with average ambient temperature of 58°.13 2.13 Davidson (“Update on Ozone Trends in California’s South Coast Air Basin,” Air and Waste, 43, 226, 1993) studied the ozone levels in the South Coast Air Basin of California for the years 1976–1991. He believes that the number of days the ozone levels exceeded 0.20 ppm (the response) depends on the seasonal meteorological index, which is the seasonal average 850-millibar temperature (the regressor). The following table gives the data.YearDaysIndex19769116.7197710517.1197810618.2197910818.119808817.219819118.219825816.019838217.219848118.019856517.219866116.919874817.119886118.219894317.319903317.519913616.6a. Make a scatterplot of the data.b. Estimate the prediction equation.c. Test for significance of regression.d. Calculate and plot the 95% confidence and prediction bands.
14 2.14 Hsuie, Ma, and Tsai (“Separation and Characterizations of Thermotropic Copolyesters of p-Hydroxybenzoic Acid, Sebacic Acid, and Hydroquinone,” Journal of Applied Polymer Science, 56, 471–476, 1995) study the effect of the molar ratio of sebacic acid (the regressor) on the intrinsic viscosity of copolyesters (the response). The following table gives the data.RatioViscosity1.00.450.90.200.80.340.70.580.60.700.50.570.40.550.30.44a. Make a scatterplot of the data.b. Estimate the prediction equation.c. Perform a complete, appropriate analysis (statistical tests, calculation of R2, and so forth).d. Calculate and plot the 95% confidence and prediction bands.
15 2.15 Byers and Williams (“Viscosities of Binary and Ternary Mixtures of Polynomatic Hydrocarbons,” Journal of Chemical and Engineering Data, 32, 349–354, 1987) studied the impact of temperature on the viscosity of toluene–tetralin blends. The following table gives the data for blends with a 0.4 molar fraction of toluene.Temperature (°C)Viscosity (mPa · s)24.91.133035.00.977244.90.853255.10.755065.20.672375.20.602185.20.542095.20.5074a. Estimate the prediction equation.b. Perform a complete analysis of the model.c. Calculate and plot the 95% confidence and prediction bands.
16 2.16 Carroll and Spiegelman (“The Effects of Ignoring Small Measurement Errors in Precision Instrument Calibration,” Journal of Quality Technology, 18, 170–173, 1986) look at the relationship between the pressure in a tank and the volume of liquid. The following table gives the data. Use an appropriate statistical software package to perform an analysis of these data. Comment on the output produced by the software routine.VolumePressureVolume20844599284220844600303022735044303122735043303122735044322124635488322124635487340926515931341026525932360026525932360028426380378863803789859968183789860068173979904868183979904872664167948472684168948777094168948777104358993681564358993881584546103778597454710379
17 2.17 Atkinson (Plots, Transformations, and Regression, Clarendon Press, Oxford, 1985) presents the following data on the boiling point of water (°F) and barometric pressure (inches of mercury). Construct a scatterplot of the data and propose a model that relates boiling point to barometric pressure. Fit the model to the data and perform a complete analysis of the model using the techniques we have discussed in this chapter.Boiling PointBarometric Pressure199.520.79199.320.79197.922.40198.422.67199.423.15199.923.35200.923.89201.123.99201.924.02201.324.01203.625.14204.626.57209.528.49208.627.76210.729.64211.929.88212.230.06
18 2.18 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 commercial 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 regressor is the amount of money spent by the firm on advertising. The data follow.FirmAmount Spent (millions)Returned Impressions per week (millions)Miller Lite50.132.1Pepsi74.199.6Stroh’s19.311.7Federal Express22.921.9Burger King82.460.8Coca-Cola40.178.6McDonald’s185.992.4MCI26.950.7Diet Cola20.421.4Ford166.240.1Levi’s2740.8Bud Lite45.610.4ATT Bell154.988.9Calvin Klein512Wendy’s49.729.2Polaroid26.938Shasta5.710Meow Mix7.612.3Oscar Meyer9.223.4Crest32.471.1Kibbles N Bits6.14.4a. Fit the simple linear regression model to these data.b. Is there a significant relationship between the amount a company spends on advertising and retained impressions? Justify your answer statistically.c. Construct the 95% confidence and prediction bands for these data.d. Give the 95% confidence and prediction intervals for the number of retained impressions for MCI.
19 2.19 Table B.17 Contains the Patient Satisfaction data used in Section 2.7.a. Fit a simple linear regression model relating satisfaction to age.b. Compare this model to the fit in Section 2.7 relating patient satisfaction to severity.
20 2.20 Consider the fuel consumption data given in Table B.18. The automotive engineer believes that the initial boiling point of the fuel controls the fuel consumption. Perform a thorough analysis of these data. Do the data support the engineer’s belief?
21 2.21 Consider the wine quality of young red wines data in Table B.19. The winemakers believe that the sulfur content has a negative impact on the taste (thus, the overall quality) of the wine. Perform a thorough analysis of these data. Do the data support the winemakers’ belief?
22 2.22 Consider the methanol oxidation data in Table B.20. The chemist believes that ratio of inlet oxygen to the inlet methanol controls the conversion process. Perform a through analysis of these data. Do the data support the chemist’s belief?
23 2.23 Consider the simple linear regression model y = 50 + 10x + ε where ε is NID (0, 16). Suppose that n = 20 pairs of observations are used to fit this model. Generate 500 samples of 20 observations, drawing one observation for each level of x = 1, 1.5, 2, …, 10 for each sample.a. For each sample compute the least-squares estimates of the slope and intercept. Construct histograms of the sample values of and . Discuss the shape of these histograms.b. For each sample, compute an estimate of E(y|x = 5). Construct a histogram of the estimates you obtained. Discuss the shape of the histogram.c. For each sample, compute a 95% CI on the slope. How many of these intervals contain the true value β1 = 10? Is this what you would expect?d. For each estimate of E(y|x = 5) in part b, compute the 95% CI. How many of these intervals contain the true value of E(y|x = 5) = 100? Is this what you would expect?
24 2.24 Repeat Problem 2.23 using only 10 observations in each sample, drawing one observation from each level x = 1, 2, 3, …, 10. What impact does using n = 10 have on the questions asked in Problem 2.23? Compare the lengths of the CIs and the appearance of the histograms.
25 2.25 Consider the simple linear regression model y = β0 + β1x + ε, with E(ε) = 0, Var(ε) = σ2, and ε uncorrelated.a. Show that .b. Show that .
26 2.26 Consider the simple linear regression model y = β0 + β1x + ε, with E(ε) = 0, Var(ε) = σ2, and ε uncorrelated.a. Show that .b. Show that E(MSRes) = σ2.
27 2.27 Suppose that we have fit the straight-line regression model but the response is affected by a second variable x2 such that the true regression function isa. Is the least-squares estimator of the slope in the original simple linear regression model unbiased?b. Show the bias in .
28 2.28 Consider the maximum-likelihood estimator of σ2 in the simple linear regression model. We know that is a biased estimator for σ2.a. Show the amount of bias in .b. What happens to the bias as the sample size n becomes large?
29 2.29 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 observations x1, x2, …, xn be taken? Discuss the practical aspects of this data collection plan.
30 2.30 Consider the data in Problem 2.12 and assume that steam usage and average temperature are jointly normally distributed.a. Find the correlation between steam usage and monthly average ambient temperature.b. Test the hypothesis that ρ = 0.c. Test the hypothesis that ρ = 0.5.d. Find
