1. (3 points) The following is part of an ANOVA table from a regression of Y on a single X where n = 100.
Source DF Sum of Squares Mean Square F-ratio
Model
Error
Total
5566
56
a) Fill in the seven blanks in the table above.
b) Calculate R2 for these data.
c) Calculate adjusted R2 for these data.
d) Calculate sY |X, the conditional standard deviation of Y given X.
e) Calculate sY , the marginal standard deviation of Y .
f) Calculate the t-statistic for testing the null hypothesis H0 : β1 = 0.
2. Data Analysis of Gas Mileage (5 points)
Variation in gasoline mileage among makes and models of automobiles is influenced by many different variables. The EPA keeps yearly data on cars. Data from 2022 are on Moodle. We want to estimate a regression of MPG using the variables Pass Vol, Eng size, and Num Cyl as predictors.
Each line records a Make/Model of a car. The data structure is as follows:
• Year: Model year
• MfrName: Manufacturer
• Model: model of the car
• MPG: Average miles per gallon as per the EPA
• PassVol: Average interior passenger volume of the car
• EngSize: Engine size of the car
• NumCyl: Number of cylinders
• Cyl4: Indicator variable that equals 1 when Num Cyl = 4
• Cyl6: Indicator variable that equals 1 when Num Cyl = 6
• Cyl8: Indicator variable that equals 1 when Num Cyl = 8
• Cyl12: Indicator variable that equals 1 when Num Cyl = 12
a) Estimate the following regression
MPGi = β0 + β1 Pass Voli + β2 Eng sizei + β3 Cyl6i + β4 Cyl8i + β5 Cyl12i + εi
Write out the four simplified prediction equations for this complicated analy- sis of covariance.
b) Calculate the VIFjs for each of the predictors. (Show your work–but feel free to use R to get R2 for each predictor and to check your answers) Are you concerned about multicollinearity in this dataset at all?
c) Calculate 95% Confidence Intervals of each of the regression coefficients–show me your calculations.
d) Using R, calculate both a 95% individual prediction interval and a 95% mean confidence interval for a new car that has a PassVol = 100, an EngSize = 5.0, and a NumCyl = 6. Write interpretations of these interval.
e) Look at the residual plots of your model. Does it indicate any violation of the assumptions of linearity, constant variance or normality?
3. (3 points) The following graphs are used to verify some of the assumptions of the ordinary least squares regression of Y on X1, X2,...Xp.
1.) The scatter plot matrix of the variables X1, X2,...Xp 2.) The normal probability plot (QQ-plot) of the residuals 3.) The residuals versus fitted (or predicted) values
For each of these graphs:
a) What assumption can be verified or shown to be violated by the graph?
b) Draw an example of the graph where the assumption does not seem to be violated.
c) Draw an example of the graph which indicates the violation of the assumption.
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