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STAT 350 (Spring 2017) Homework 12 (20 points + 9 points BONUS) 1

Practice Problems: 12.35 (p.601), 12.37 (p.601), 12.39 (p.602) (1 pt. BONUS) 1. Given that:

𝑏1 = ∑[(𝑥𝑖 − �̅�)(𝑦𝑖 − �̅�)]

∑(𝑥𝑖 − �̅�) 2

= ∑ 𝑎𝑖 𝑦𝑖

Show that

√∑ 𝑎𝑖 2 =

1

√𝑆𝑋𝑋 𝑤ℎ𝑒𝑟𝑒 𝑆𝑋𝑋 = ∑(𝑥𝑖 − �̅�)

2

(9 pts.) 2. Many factors affect the length of a professional football game. A study was conducted

to determine the relationship between the total number of penalty yards (x) and the time required to complete a game (y, in hours). You may assume that all of the assumptions for regression are valid. The summary data is given below:

n = 9, SXX = 26 256, SYY = 3.956, SXY = 244.8, MSE = 0.239 The line is: ŷ = 2.322 + 0.00932 x (2 pts.) a) Find and interpret the 95% confidence interval for the slope. (6 pts.) b) Conduct a hypothesis test (t-test) to determine if there is an association between the

time of the game and the total number of penalty yards. Please use a significance level of 0.05.

(1 pt.) c) Find the sample correlation coefficient. (2 pts.) 3. Golden Rule medical insurance company recently investigated the relationship

between the number of patients per registered nurse in a hospital and the patient’s length of stay. A random sample of hospitals was selected, and the number of patients per registered nurse was computed (x). A patient was randomly selected for each hospital, and the length of stay was recorded (y, in hours). Assume that the assumptions for linear regression are met. The scatterplot of length of stay (y) vs. the number of patients per registered nurse at that hospital (x) is shown below:

(1 pt.) a) Would it be appropriate to use this study to predict average length of stay of a patient

at a hospital where the average number of patients per registered nurse is 3.35? Please explain your answer.

(1 pt.) b) Would it be appropriate to use this study to predict average length of stay of a patient at a hospital where the average number of patients per registered nurse is 5.15? Please explain your answer.

STAT 350 (Spring 2017) Homework 12 (20 points + 9 points BONUS) 2

Additional Problems: 12.41, 12.43, 12.45, 12.51, Practice Problems: 12.93 (p.620) (3 pts.) 4. For each of the following residual plots, indicate whether the graph violates each of

the following assumptions: 1) linearity and 2) constant variance. (1 pt.) a) (1 pt.) b)

(1 pt.) c)

Additional Problems: 12.97 Practice Problems: 12.67 (p. 611)

STAT 350 (Spring 2017) Homework 12 (20 points + 9 points BONUS) 3

(5 pts.) 5. Highway engineers have long argued that roads designed with high skid resistance help to prevent accidents, especially in wet conditions. A random sample of two-lane highways was selected from across the United States, and the skid resistance was measured (in skid numbers) using a Skid Resistance Tester (SRT). The accident rate (per 10,000 vehicles) was computed for 25-mile sections of each highway during wet conditions. The following summary statistics were reported.

n = 20 b0 = 1.157 b1 = -1.228 x̄ = 0.5165 MSE = 0.04570 SXX = 0.9475 (2 pts.) a) Find and interpret a 95% confidence interval for the population accident rate when the

skid resistance is 0.50. (2 pts.) b) Find and interpret a 95% prediction interval for the accident rate when the skid

resistance is 0.50. (1 pt.) c) Without doing any calculations, do you think that the population accident rate is larger

when the skid resistance is 0.60 then when it is 0.50? Please explain your answer. Additional Problems: 12.75, 12.77 Practice Problems: 12.125 (p.637) (1 pt.) 6. Is the following model appropriate for linear regression? Please explain your answer

by listing each term and indicating whether it is appropriate or not.

Y = 0 + 1x1 + 2e x2 + 𝛽3𝑥3

𝛽4 + ε

STAT 350 (Spring 2017) Homework 12 (20 points + 9 points BONUS) 4

(8 pts.) 7. BONUS A recent study investigated the variables that can be used to predict economic growth in China, as measured by gross domestic product (GDP, y, in hundred millions). A sample of years was obtained and the following measurements were recorded: capital stock (x1, in hundred millions); labor force (x2, in ten thousands); and energy consumption (x3, in ten thousand tons standard coal). The following are the results from SAS:

(1 pt.) a) What is the formula for the estimated regression line? (6 pts.) b) Is the overall regression significant at a significance level of 0.01? Hint: Perform F test

at a significance level of 0.01. (1 pt.) c) Which of the variables are linearly associated with GDP assuming that the rest of the

variables are in the model? You may assume a significance level of 0.01. You do not have to perform any significance tests but you have to explain how you arrive at your answer.

Additional Problems: 12.143