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A collector of antique grandfather clocks believes that the price (in dollars) received for the clocks at an antique auction increases with the age of the clocks and with the number of bidders. Thus the model is hypothesized is where Y = auction price, x1 = age of clock (years) and x2 = number of bidders.

A sample of 32 auction prices of grandfather clocks, along with their ages and the number of bidders, is given below.

Age (x1) Bidders (x2) Price (y) Age (x1) Bidders (x2) Price (y)
127 13 1235 170 14 2131
115 12 1080 182 8 1550
127 7 845 162 11 1884
150 9 1522 184 10 2041
156 6 1047 143 6 854
182 11 1979 159 9 1483
156 12 1822 108 14 1055
132 10 1253 175 8 1545
137 9 1297 108 6 729
113 9 946 179 9 1792
137 15 1713 111 15 1175
117 11 1024 187 8 1593
137 8 1147 111 7 785
153 6 1092 115 7 744
117 13 1152 194 5 1356
126 10 1336 168 7 1262

a) State the multiple regression equation.

b) Interpret the meaning of the slopes b1 and b2 in the model.

c) Interpret the meaning of the regression coefficient b0.

d) Test H0: ?2 = 0 against H1: ?2 > 0. Interpret your finding.

e) Use a 95% confidence interval to estimate ?2. Interpret the p-value corresponding to the estimate ?2. Does the confidence interval support your interpretation in d)?

f) Determine the coefficient of multiple determination r2Y.12 and interpret its meaning.

g) Perform a residual analysis on your results and determine the adequacy of the fit of the model.

h) Plot the residuals against the prices. Is there evidence of a pattern in the residuals? Explain.

i) At ? = 0.05, is there evidence of positive autocorrelation in the residuals?

j) Suppose the collector, having observed many auctions, believes that the rate of increase of the auction price with age will be driven upward by a large number of bidders. In other words, the collector believes that the age of clock and the number of bidders should interact. Is there evidence to support his claim that the rate of change in the mean price of the clocks with age increases as the number of bidders increases? Should the interaction term (x1 x2) be included in the model? If so, what is the multiple regression equation?

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This solution answers questions regarding the computer output from a multiple regression analysis.

Assume a significance level of 5%

1. Everybody seems to disagree about why so many parts have to be fixed or thrown away after they are produced. Some say that it's the temperature of the production process, which needs to be held constant (within a reasonable range). Others claim that it's clearly the density of the product, and that if we could only produce a heavier material, the problems would disappear. Then there is Ole, who has been warning everyone forever to take care not to push the equipment beyond its limits. This problem would be the easiest to fix, simply by slowing down the production rate; however, this would increase costs. Interestingly, many of the workers on the morning shift think that the problem is "those inexperienced workers in the afternoon," who, curiously, feel the same way about the morning workers.

Ever since the factory was automated, with computer network communication and bar code readers at each station, data have been piling up. You've finally decided to have a look. After your assistant aggregated the data by 4-hour blocks and then typed in the Morning variable, you found the following note on your desk with a printout of the data already loaded into the computer network:

 Temperature actually measures temperature variability as a standard deviation
during the time of measurement. Units are degrees Fahrenheit.

 Density indicates the density of the final product. Units are ounces per cubic inch.

 Rate indicates the rate of production. Units per hour.

 Morning is an indicator variable that is 1 during morning production and is 0 during the afternoon.

 Defect is the number of defects per 1,000 produced.

You decide to run a regression to determine the effect of the variables Temperature, Density, Rate, and Morning on the number of defects. Use the output on the next page to answer the following questions. Each question is worth 3 pts.

a) Report the R2 value and explain its interpretation (in non-technical language).

b) Explain in non-technical language the coefficient of the variable Temperature. (Be careful here.)

Results of multiple regression for Defect

(see attached file for table)

c) What does the p-value on Temperature indicate?

d) What does Temperature's significance or lack thereof imply about controlling the production quality? In particular, should you be looking to control temperature within a reasonable range?

e) Explain in non-technical language the coefficient of the indicator variable Morning. In particular, which shift (the morning or afternoon) is producing the most defects?

f) What does Morning's significance or lack thereof imply about controlling the production quality? What action does the p-value on Morning indicate you should take?

g) Even though you feel the above results are good, you remember you statistics teacher going on and on about the importance of looking at summary measures and graphing the variables. Reluctantly, you decide to view the summary stats:

Summary measures for selected variables
Temperature Density Rate Morning Defect
Count 30 30 30 30 30
Mean 2.2 25.3 236.5 0.800 27.1
Standard dev. 0.6 3.4 26.1 1.808 19.4
Minimum 1.0 19.5 177.7 0 0.0
Maximum 3.0 32.2 281.9 10 60.8
Range 2.1 12.7 104.2 10 60.8

After close inspection, you very thankful that you listened to your statistics teacher! Why?

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