Introduction to Linear Regression Analysis. Douglas C. Montgomery

and c) using vehicle weight x10 as the regressor variable. Based on a comparison of the two models, can you conclude that x1 is a better choice of regressor than x10?

      6 2.6 Table B.4 presents data for 27 houses sold in Erie, Pennsylvania.a. Fit a simple linear regression model relating selling price of the house to the current taxes (x1).b. Test for significance of regression.c. What percent of the total variability in selling price is explained by this model?d. Find a 95% CI on β1.e. Find a 95% CI on the mean selling price of a house for which the current taxes are $750.

      7 2.7 The purity of oxygen produced by a fractional distillation process is thought to be related to the percentage of hydrocarbons in the main condensor of the processing unit. Twenty samples are shown below.Purity (%)Hydrocarbon (%)86.911.0289.851.1190.281.4386.341.1192.581.0187.330.9586.291.1191.860.8795.611.4389.861.0296.731.4699.421.5598.661.5596.071.5593.651.4087.311.1595.001.0196.850.9985.200.9590.560.98a. Fit a simple linear regression model to the data.b. Test the hypothesis H0: β1 = 0.c. Calculate R2.d. Find a 95% CI on the slope.e. Find a 95% CI on the mean purity when the hydrocarbon percentage is 1.00.

      8 2.8 Consider the oxygen plant data in Problem 2.7 and assume that purity and hydrocarbon percentage are jointly normally distributed random variables.a. What is the correlation between oxygen purity and hydrocarbon percentage?b. Test the hypothesis that ρ = 0.c. Construct a 95% CI for ρ.

      9 2.9 Consider the soft drink delivery time data in Table 2.10. After examining the original regression model (Example 2.9), one analyst claimed that the model was invalid because the intercept was not zero. He argued that if zero cases were delivered, the time to stock and service the machine would be zero, and the straight-line model should go through the origin. What would you say in response to his comments? Fit a no-intercept model to these data and determine which model is superior.

      10 2.10 The weight and systolic blood pressure of 26 randomly selected males in the age group 25–30 are shown below. Assume that weight and blood pressure (BP) are jointly normally distributed.a. Find a regression line relating systolic blood pressure to weight.b. Estimate the correlation coefficient.c. Test the hypothesis that ρ = 0.d. Test the hypothesis that ρ = 0.6.e. Find a 95% CI for ρ.SubjectWeightSystolic BP1165130216713331801504155128521215161751467190150821014092001481014912511158133121691351317015014172153151591281616813217174149181831581921515020195163211801562214312423240170242351652519216026187159

      11 2.11 Consider the weight and blood pressure data in Problem 2.10. Fit a no-intercept model to the data and compare it to the model obtained in Problem 2.10. Which model would you conclude is superior?

      12 2.12 The number of pounds of steam used per month at a plant is thought to be related to the average monthly ambient temperature. The past year’s usages and temperatures follow.MonthTemperatureUsage/l000Jan.21185.79Feb.24214.47Mar.32288.03Apr.47424.84May50454.68Jun.59539.03Jul.68621.55Aug.74675.06Sep.62562.03Oct.50452.93Nov.41369.95Dec.30273.98a. Fit a simple linear regression model to the data.b. Test for significance of regression.c. Plant management believes that an increase in average ambient temperature of 1 degree will increase average monthly steam consumption by 10,000 lb. Do 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