exam1at430sol

STAT 425 Exam 1 @ 4:30 pm October 4, 2023, 4:30pm Name: SOLUTIONS Netid: _________________________ This is an 80 minute handwritten exam. There are 5 problems, each worth 10 points. Do not start working until your proctor tells you to start. Your head must be visible to your proctor on Zoom along with your screen and the work in front of you. You may use a calculator but not the computer or R or the internet for your work. To work on the exam you can either: 1. Print out the exam and do your handwritten work on the exam itself; or 2. View the exam on your screen and do work on separate sheets of paper. Clearly label your work as to which problem number (1,2,3..) and part (a,b,c..) you are solving; or 3. Do work on a blank file or pdf of the exam on a tablet, and upload your work file from the tablet. In this case the proctor must be able to see your tablet. Scanning and uploading your exam: After you finish, scan/photograph each page and upload into Moodle in the same way you upload assignment files. You are allowed two one-sided 8.5 by 11 inch sheets of notes for yourself. Scan and upload after the exam. 1

Problem 1. (3 parts) Data will be collected in the form ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x n , y n ) , where x i is the i th value of a fixed, nonrandom explanatory variable, and y i is the corresponding random response. Consider the model y i = β 0 + β 1 x i + e i , i = 1 , . . . , n for unknown parameters β 0 and β 1 and random errors e 1 , . . . , e n that are independent with mean zero and variances equal to an unknown constant σ 2 . The ordinary least squares estimators for β 0 and β 1 are given by ˆ β 0 = ¯ y − ¯ x ˆ β 1 and ˆ β 1 = QQQQQQQ n i =1 ( x i − ¯ x )( y i − ¯ y ) QQQQQQQ n i =1 ( x i − ¯ x ) 2 , where ¯ x = 1 n n YYYYYYY i =1 x i and ¯ y = 1 n n YYYYYYY i =1 y i . (a) (3 pts) If x 1 = 3 . 5 and x 2 = 5 . 0 , find E ( y 2 ) − E ( y 1 ) in terms of the model parameters. E ( y 2 ) − E ( y 1 ) = ( β 0 + 5 β 1 ) − ( β 0 + 3 . 5 β 1 ) = 1 . 5 β 1 (b) (3 pts) Find an explicit expression for E ( ¯ y ) in terms of the model parameters and predictor variables. E (¯ y ) = E 1 n n YYYYYYY i =1 y i = 1 n n YYYYYYY i =1 E ( y i ) = 1 n n YYYYYYY i =1 ( β 0 + β 1 x i ) = β 0 + β 1 1 n n YYYYYYY i =1 x i = β 0 + β 1 ¯ x (c) (4 pts) After the data are collected we find n = 20 , QQQQQQQ 20 i =1 ( x i − ¯ x ) 2 = 50 , QQQQQQQ 20 i =1 ( y i − ¯ y ) 2 = 33 , and QQQQQQQ 20 i =1 ( y i − ˆ y i ) 2 = 9 . 6 , where ˆ y 1 , . . . , ˆ y 20 are the fitted values for LS regression of y on x . Based on these results, show how to calculate the standard error for ˆ β 1 , plugging in all the relevant numbers. You do not have to complete the calculation. se ( ˆ β 1 ) = ˆ σ rrrrrrr QQQQQQQ 20 i =1 ( x i − ¯ x ) 2 = wwwwwww vvvvvvv vvvvvvv uuuuuuu QQQQQQQ 20 i =1 ( y i − ˆ y i ) 2 / (20 − 2) QQQQQQQ 20 i =1 ( x i − ¯ x ) 2 = ttttttt 9 . 6 / 18 50 2

Problem 2. (4 parts) Data on fuel consumption were collected for each of the 50 states and Washington D.C. for a total sample size of n = 51 . The variables are gasoline Tax (cents/gallon), Fuel consumption per 1000 pop. over 16, Dlic (Licensed Drivers per 1000 population over 16), and logMiles (log 10 miles of highway in the state). The following linear model was fit to the data. Fuel = β 0 + β 1 Tax + β 2 Dlic + β 3 logMiles + error The results of fitting a linear model of this form are summarized below. ## ## Call: ## lm(formula = Fuel ~ Tax + Dlic + logMiles, data = df) ## ## Residuals: ## Min 1Q Median 3Q Max ## -171.13 -48.91 5.34 41.90 193.25 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -166.926 168.544 -0.99 0.32705 ## Tax -3.999 2.171 -1.84 0.07175 ## Dlic 0.536 0.135 3.96 0.00025 ## logMiles 79.445 21.972 3.62 0.00073 ## ## Residual standard error: 69.4 on 47 degrees of freedom ## Multiple R-squared: 0.427, Adjusted R-squared: 0.391 ## F-statistic: 11.7 on 3 and 47 DF, p-value: 7.66e-06 (a) (2 pts) Based on the results, what is the proportion of total variance explained by the model? Multiple R-squared = 0.427 (b) (2 pts) Based on the fitted model, estimate the expected Fuel consumption per 1000 population for a state with the following profile: ## Tax Dlic logMiles ## 20 782.8 5.1 Set up the calculation with all relevant numbers. You do not need to complete the calculation. -166.926 + (20)(-3.999) + (782.8)(0.536) + (5.1)(79.445) 3