CSCI E-106:Assignment 3 Solved

CSCI E-106:Assignment 3
Instructions

Students should submit their reports on Canvas. The report needs to clearly state what question is being solved, step-by-step walk-through solutions, and final answers clearly indicated. Please solve by hand where appropriate.

Please submit two files: (1) a R Markdown file (.Rmd extension) and (2) a PDF document, word, or html generated using knitr for the .Rmd file submitted in (1) where appropriate. Please, use RStudio Cloud for your solutions.



Problem 1
Five observations on Y are to be taken when X = 4, 8, 12, 16, and 20, respectively. The true regression function is E(Y} = 20 + 4X, and the i are independent N(O, 25). (40 points, 10 points each)

a-) Generate five normal random numbers, with mean 0 and variance 25. Consider these random numbers as the error terms for the five Y observations at X = 4,8, 12, 16, and 20 and calculate Y1, Y2, Y3, Y4 , and Y5. Obtain the least squares estimates b0 and b1, when fitting a straight line to the five cases. Also calculate Yh when Xh = 10 and obtain a 95 percent confidence interval for E(Yh) when Xh = 10. b-) Repeat part (a) 200 times, generating new random numbers each time.

c-) Make a frequency distribution of the 200 estimates b1. Calculate the mean and standard deviation of the 200 estimates b1. Are the results consistent with theoretical expectations?

d-) What proportion of the 200 confidence intervals for E(Yh) when Xh = 10 include E(Yh)? Is this result consistent with theoretical expectations?

Problem 2
Refer to the CDI data set (used in homework 1). The number of active physicians in a CDI (Y) is expected to be related to total population, number of hospital beds, and total personal income. Using R2 as the criterion, which predictor variable accounts for the largest reduction in the variability in the number of active physicians? (20 Point)

Problem 3
Refer to the CDI data set (use in previous homework). For each geographic region, regress per capita income in a CDI (Y) against the percentage of individuals in a county having at least a bachelor’s degree (X). Obtain a separate interval estimate of β1, for each region. Use a 90 percent confidence coefficient in each case. Do the regression lines for the different regions appear to have similar slopes? (20 points)

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Problem 4
In a small-scale regression study, five observatiol)s on Y were obtained corresponding to X = 1, 4,10, ll, and

14. Assume that σ = .6, β0 = 5, and β1, = 3. (20 points, 10 points each) a-) What are the expected values MSR and MSE?

b-) For determining whether or not a regression relation exists, would it have been better or worse to have made the five observations at X = 6,7, 8, 9, and 1O? Why? Would the same answer apply if the principal purpose were to estimate the mean response for X = 8? Discuss.

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