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Statistics 328 Examination 2 Name Fall 1996 You must show all of your work
When asked to explain something, or to provide an interpretation for a quantity, provide an explanation that could be understood by someone who does not have formal training in statistical methods.
- Data were collected on the auction sale prices of 32 antique clocks. For each observation in the data set, the price in dollars (Y ), the age of the clock in years (x 1 ), and the number of bidders (x 2 ) was recorded. The following figure is a “pairs plot” showing all pair-wise scatter plots.
Age
6 8 10 12 14
120
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6
8 10
12
(^14) •
Bidders
120 140 160 180
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800 1200 1600 2000 800
1200
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2000 Price
Three models were fit to the data
Model 1: y = β 0 + β 1 x 1 + β 2 x 2 + Model 2: y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 1 x 2 + Model 3: y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 1 x 2 + β 4 x^21 + β 5 x^22 +
where ∼ nid(0, σ^2 ). The following tables gives a summary of the results: Model 1 Model 2 Model 3 β 0 − 1336. 7221 322. 7544 − 340. 033 β 1 12. 7362 0. 8733 3. 4144 β 2 85. 8151 − 93. 4099 13. 6289 β 3 1. 2979 1. 1234 β 4 − 0. 0037 β 5 1. 1234 SSE =
i=1(yi^ −^ ŷi)
SSyy =
i=1(yi^ −^ ¯y)
(a) List some of the things that you see in the pairs plot.
(b) Briefly explain why
1 (yi^ −^ y¯) (^2) is the same for all three models.
(c) What is an estimate for the change in expected price of an auctioned clock for an additional year of age for Model 1? For Model 2? Compute a numerical value if possible. If you cannot compute a numerical value, give a formula and explain how you would do the computation if you had all of the needed information.
(d) One of the assumptions used in the fitted models for the clock data is that the residuals have a standard deviation σ that does not depend on the levels of the explanatory variables (i.e., on age or the number of bidders). The residual (or error) degrees of freedom can be viewed as the effective sample size for estimating σ. Compute the residual degrees of freedom for Model 1, Model 2, and for Model 3.
(e) Compute an estimate of σ from Model 2. Briefly explain the practical interpretation of this quantity.
(f) Model 2 has an interaction term and Model 1 does not. Briefly explain, in terms of this partic- ular example, the practical difference between Model 1 and Model 2.
(g) Use an F -test to compare Model 1 and Model 2. Use α = .05. What do you conclude?
(h) Model 3 includes quadratic terms. Do the data provide sufficient evidence to include these terms in the model? Perform the appropriate test, using α = .05.
(i) Normal probability plots (and the closely related normal Q-Q plots) are useful tools for data analysis. The following figure is a normal probability plot for Model 2 of the Clock data.
- A test was conducted to compare the life times of two different light bulbs designs—the current design, and a new design. Ten bulbs for each brand were tested until failure with the number of hours until failure being recorded for each bulb. A dummy variable regression model was used to analyze the data. The model fitted is
Hours = β 0 + β 1 X 1 +
with X1 = 0 for the old design and X1 = 1 for the new design. Management asked for the life test so that they could decide whether to switch production to the new design or not. The following is some of the Splus output for these data.
attach(bulb.frame) bulb.fit1 <- lm(Hours ~ X1) summary(bulb.fit1) Call: lm(formula = Hours ~ X1) Residuals: Min 1Q Median 3Q Max -271.3 -112.5 -14.6 119.5 218. Coefficients: Value Std. Error t value Pr(>|t|) (Intercept) 951.9000 45.8659 20.7540 0. X1 146.4000 64.8642 2.2570 0. Residual standard error: 145 on 18 degrees of freedom Multiple R-Squared: 0. F-statistic: 5.094 on 1 and 18 degrees of freedom, the p-value is 0. anova(bulb.fit1) Analysis of Variance Table Response: Hours Terms added sequentially (first to last) Df Sum of Sq Mean Sq F Value Pr(F) X1 1 107164.8 107164.8 5.094151 0. Residuals 18 378663.0 21036. other.X1 <- list(X1=c(0,1)) predict(bulb.fit1,newdata=other.X1,se.fit=T) $fit: 1 2 951.9 1098.
$se.fit: 1 2 45.86593 45. $residual.scale: [1] 145. $df: [1] 18
(a) Compute a confidence interval for the mean life for the old design. What is the interpretation of this interval?
(b) Compute a confidence interval for the difference in mean life between the new and the old designs.
(c) Compute a prediction interval for the life of a light bulb from the new design.
(d) Is there evidence that the mean life of the new design is longer than that of the old design?
