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1 STAT 351 Final Exam - Summer 2020 100 Points This is a take home exam and it must be done individually. You are not allowed to discuss this exam with others. • Due date: Friday, June 26, 2020 at 6.00 p.m. CDT. You are given 36 hours to write your answers and submit your exam through one of the three methods given below. • Submission direction: Submit your answers via one of the following methods. Please take extra care on submission through handwritten and snapshots (images) methods, that your images are easily readable. 1) Type your answers in a word document and submit as a word document file or a pdf file. 2) Write down your answers on separate sheets of paper and submit a scanned copy of the answers. 3) Write down your answers on separate sheets of paper and submit snapshots of the answers. If you are submitting multiple images, please name each image with the corresponding question number and corresponding parts of the questions (Ex: an image name could be Q3. a, b, c). • Note: 1. Please make sure that the images are CLEAR enough to read with good brightness and contrast. 2. If your answers are handwritten, they must be LEGIBLE/READBLE. 3. Write your answers in order as given in the exam. • Warning: 1. Answers with non-readable handwriting and poor snapshot images will lower your grades. So, make use of the time to carefully prepare your exam. • Direction: 1. To answer questions, please write your answers with question number and corresponding parts of the questions. Ex: Q3. c 2. Write your answers in order as given in the exam. • Technical Support: If you have any technical difficulties with accessing exam, uploading answers and other exam related problems, please use my contact information and I will respond as quicker as possible. Email:
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[email protected] 2 All tests of hypotheses are to be done at the 5% level of significance (? = ?. ??) and all confidence intervals are to be 95% unless otherwise stated. 1. The following information regarding a dependent variable Y and an independent variable X is provided. Answer part (a) to (g) using the following information. �̅� = 11 ∑(?? − �̅�)(?? − �̅�) = 360 5 ?=1 �̅� = 30 ∑(?? − �̅�) 2 5 ?=1 = 90 ? = 5 ??? = 1850 ??? = 90 ??1 = 0.577 (a) 4pts. Find the slope of the estimated regression equation (b1). (b) 4pts. Find the intercept of the estimated regression equation (b0). (c) 4pts. Develop the least squares estimated regression equation. (d) 5pts. Find the point estimate of ?(?) when ? = 4. (e) 10pts. Carry out a t-test to see whether the predictor(x) is significant. Use α = 0.05 and the critical value approach to draw your conclusion. Make sure to show all your steps. State your conclusion in a complete sentence and use words a non- statistician would understand. (f) 4pts. Find the coefficient of determination (r2). (g) 4pts. Please interpret the coefficient of determination (r2). (h) 2pts. Find coefficient of correlation (rxy). 2. The following Regression function has been developed to check the relationship between Y- ‘Sales’ and the following independent variables; X1- Time (Length of time employed in months) X2 –Poten (Market potential) X3 – AdvExp (Advertising expenditure in the sales territory) X4 – Accounts (Number of accounts assigned to sales rep) 3 Consider the following Minitab output and answer the questions. Regression Analysis: Sales versus Time, Poten, AdvExp, Accounts Regression Equation Sales = -391 - 0.58 Time + 0.02227 Poten + 0.2178 AdvExp + 16.31 Accounts Coefficients Term Coef SE Coef T-Value P-Value VIF Constant -391 470 -0.83 0.415 Time -0.58 2.20 -0.26 0.794 2.46 Poten 0.02227 0.00895 2.49 0.022 1.34 AdvExp 0.2178 0.0468 4.65 0.000 1.07 Accounts 16.31 4.20 3.88 0.001 2.48 Model Summary S R-sq R-sq(adj) R-sq(pred) 595.985 82.83% 79.40% 72.74% Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 4 34275579 8568895 24.12 0.000 Time 1 24781 24781 0.07 0.794 Poten 1 2198350 2198350 6.19 0.022 AdvExp 1 7683578 7683578 21.63 0.000 Accounts 1 5346156 5346156 15.05 0.001 Error 20 7103970 355198 Total 24 41379549 4 a. 4 pts. Write the hypotheses of the F-test to check the overall relationship between Y and X1,X2,X3 ,X4. b. 5 pts. What is your conclusion to part (a)? (Use α =0.05). You need to justify your answer using an appropriate value from the output. Please state your conclusion in a complete sentence and use words a non-statistician would understand. c. 4 pts. Write down the hypotheses of the t-test for the independent variable, Time (X1), to test whether this independent variable and Y have a significant regression relationship. d. 4 pts. What can you say about the significance of the Time (X1 - Length of time employed in months) to the regression equation based on the corresponding t-test? (??? ? = 0.05). Please use appropriate values related to the t-test from the above output. e. 4 pts. What can you say about “goodness of fit” for the above regression model fit based on the appropriate value from the output? f. 4 pts. Is normality assumption about the error ? satisfied based on the above Minitab output? Explain. g. 4 pts. Is constant variance assumption about the error ? satisfied based on the above Minitab output? Explain. h. 6pts. Why do we use R-sq(adj) in multiple linear regression rather than R_sq? Make sure to explain in detail. 5 3. A manager in an automobile company collected data on annual sales for new customer accounts and the number of years of experience for a sample of 15 salespersons. The following is the Regression Analysis run by Minitab for developing an estimated regression equation to predict annual sales using the independent variable years of experience. Note that ? = ????? ?? ??????????, ? = ?????? ?????. Regression Analysis: Annual Sales versus Years of Experience Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 53.86 4.79 11.25 0.000 Years of Experience 8.361 0.462 18.11 0.000 1.00 Model Summary S R-sq R-sq(adj) R-sq(pred) 8.98845 96.19% 95.89% 95.17% Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 1 26491.0 26491.0 327.89 0.000 Years of Experience 1 26491.0 26491.0 327.89 0.000 Error 13 1050.3 80.8 Lack-of-Fit 11 690.3 62.8 0.35 0.901 Pure Error 2 360.0 180.0 Total 14 27541.3 Settings Variable Setting Years of Experience 8 Prediction Fit SE Fit 95% CI 95% PI 120.748 2.32101 (124.095, 134.123) (109.054, 149.165) a) 5 pts. Please write down the estimated regression equation to predict annual sales using the independent variable years of experience b) 4 pts. Find a 95% confidence interval for the mean annual sales for all salespersons with eight years of experience. c) 4 pts. The company is considering hiring Tim White, a salesperson with eight years of experience. Find a 95% prediction interval of annual sales for Tim White. d) 5 pts. Discuss the differences in your answers to part b) and c). That is, which interval estimation is wider? And why? 6 4. The following results are part of the Regression Analysis for the above question (3) from Minitab for analyzing the residuals. Here, SRES stands for standardized residual and HI for leverage values. Table 1: Residual Analysis Observation SRES HI 1 0.259783 0.42439 2 0.440919 0.280488 3 -0.46692 0.180488 4 2.257577 0.42439 5 -1.4616 0.143902 6 -0.40935 0.143902 7 1.169025 0.143902 8 -1.33051 0.219512 9 1.765634 0.439024 10 -0.40935 0.143902 11 1.169025 0.143902 12 -1.33051 0.219512 13 1.765634 0.339024 14 -1.33051 0.219512 15 1.765634 0.339024 I. 5 pts. Can any of the observations be classified as outliers? II. 5 pts. Can any of the observations be classified as influential observations?