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Solutions to practice problems related to simple linear regression analysis. The problems focus on the association between weight and mpgcity, calculating confidence intervals, interpreting the intercept, diagnosing model assumptions, and predicting mpgcity for a given weight. The document also discusses the use of log transformation to improve model assumptions.

Typology: Exams

2009/2010

1 / 2

Download Midterm 1 Solutions: Weight vs. MPGCity Regression Analysis and more Exams Statistics in PDF only on Docsity! Practice Problems, Midterm 1 Solutions 1. (a) Under the simple linear regression model, WeightWeightMPGCityE *)|( 10 . Weight is associated with Miles per Gallon in the City (MPGCity) if 01 . To test whether weight is associated with MPGCity, we test 0: 10 H versus 0: 1 aH . From the JMP output, the p-value is <0.0001. Thus, there is strong evidence that weight is associated with MPGCity. (b) 92.134000*00655.012.40)4000|(ˆ WeightMPGCityE . (c) A 95% confidence interval for the decrease in mean MPGCity when the weight is increased by one pound is approximately )007286.0,005814.0(000368.0*200655.0)ˆ(*2ˆ 11 SE . A 95% confidence interval for the decrease in mean MPGCity when the weight is increased by 1000 pounds is 1000*confidence interval for decrease in mean MPGCity when the weight is increased by one pound = )286.7,814.5())007286.0(*1000),005814.0(*1000(( . (d) The intercept technically equals the mean MPGCity when weight equals zero. But here, the mean MPGCity when weight equals zero is an extrapolation. Besides the physical impossibility of a car having zero weight, there are no cars under 1500 pounds in the data set. Here the intercept is not really interpretable. (e) (i) Narrow. (ii) Not change at all. (iii) Stay about the same. (f) The residual plot suggests that the linearity assumption is violated. The mean of the residuals for small weights of cars and large weights of cars appears to be above zero and for medium weights of cars is below zero. There is no clear violation of constant variance. The normality assumption appears to be violated because a few of the points go a little bit outside of the 95% confidence bands. To remedy the nonlinearity, I would suggest a transformation of X to X , log X or 1/X and/or a transformation of Y to Y , log Y, or 1/Y (these are the transformations suggested by Tukey’s Bulging rule. (g) From the JMP output, an approximate 95% prediction interval is (9.5, 18) miles per gallon. (h) Under the simple linear regression, the distribution of MPGCity for cars of weight 4000 pounds is normal with mean 4000*10 and standard deviation . Using the least squares estimates RMSE,ˆ,ˆ 10 of ,, 10 respectively, the distribution of MPG City given weight=4000 is approximately normal with mean 92.134000*00655.012.40)4000|(ˆ WeightMPGCityE and standard deviation 2.168, and thus

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