Statistical Foundations DEADLINE: Monday, December 14, 2020 7:30AM (EDT) You will need to do all computations in R and submit your R file. Any explanation can be added as comments in R. Make sure to...

You will need to do all computations in R and submit your R file. Any explanation can be added as comments in R. Make sure to write the final answer as a comment by using # sign. Also make sure each question is labeled clearly in your code (e.g. 1a,1b,1c…)
Only 3 problems. You have until tomorrow Monday 14th (7:30Am). That is 20 hours.


Statistical Foundations DEADLINE: Monday, December 14, 2020 7:30AM (EDT) You will need to do all computations in R and submit your R file. Any explanation can be added as comments in R. Make sure to write the final answer as a comment by using # sign. Also make sure each question is labeled clearly in your code (e.g. 1a,1b,1c…) Problem 1 (20 points total, 4 points each part) During the 1999 and 2000 baseball seasons, there was much speculation that the unusually large number of home runs that were hit was due at least in part to a livelier ball. One way to test the “liveliness” of a baseball is to launch the ball at a vertical surface with a known velocity VL and measure the ratio of the outgoing velocity VO of the ball to VL. The ratio R = VO/VL is called the coefficient of restitution. Following are measurements of the coefficient of restitution for 40 randomly selected baseballs. The balls were thrown from a pitching machine at an oak surface. 0.6248 0.6520 0.6226 0.6230 0.6237 0.6368 0.6280 0.6131 0.6118 0.6220 0.6096 0.6223 0.6159 0.6151 0.6300 0.6297 0.6298 0.6121 0.6107 0.6435 0.6192 0.6548 0.6392 0.5978 0.6351 0.6128 0.6134 0.6275 0.6403 0.6310 0.6261 0.6521 0.6065 0.6262 0.6049 0.6214 0.6262 0.6170 0.6141 0.6314 (a) Is there evidence to support the assumption that the coefficient of restitution is normally distributed? Use  = 0.01. (b) Does the data support the claim that the mean coefficient of restitution of baseballs exceeds 0.623? Use the relevant test statistic approach to support your response, assuming = 0.01. (c) What is the P-value of the test statistic computed in part (b)? (d) Compute the power of the test if the true mean coefficient of restitution is as high as 0.63. (e) What sample size would be required to detect a true mean coefficient of restitution as high as 0.63 if we wanted the power of the test to be at least 0.80? Problem 2 (15 points total, 5 points each part) We would like to check the ink production for two different production processes, say process 1 and process 2 to see whether there is a difference in the mean density of ink produced between the two processes. A sample of size 10 from both processes is taken. Process 1 has sample mean of 14.8 and a sample standard deviation of 1.4. Process 2 has sample mean of 15.6 and a sample standard deviation of 1.7. Assuming normality for both processes with same population standard deviation: (a) Perform a hypothesis test to check if both production processes produce ink with equal densities (Use the test statistic and assume  = 0.05) (b) Find the two-sided confidence interval for the difference in mean of ink densities. Can you draw the same conclusion as in part (a) from this confidence interval? (c) How does the solution for part (a) change if we assume that the two populations have different variances? Problem 3 (30 points total, 3 points each part) The electric power consumed each month by a chemical plant is thought to be related to the average ambient temperature (x1), the number of days in the month (x2), the average product purity (x3), and the tons of product produced (x4). The past year’s historical data are available and are presented in the following table: