1 Department of Mathematics and Statistics Semester 132 STAT310 Midterm Exam Name: Thursday, April 17, 2014 ID #: Question One: The director of admissions of a small college selected 120 students at random from the new freshman class in a study to determine whether a student's grade point average (GPA) at the end of the freshman year (Y) can be predicted from the ACT test score (X). The results of the study follow. provided: ∑y=, ∑x=, ∑x2 =, ∑y2 =, ∑xy= 1. Fill the missing values “?” below? The regression equation is: β1 ? Predictor Constant x Coef SE Coef T ________ ________ ?________ 0.3209 ?________ 0.01277 S = ?________ P 0.000 0.003 R-Sq = ?________ β0 Analysis of Variance Source Regression Residual Error Lack of Fit Pure Error Total DF SS ?___ ____ 34 83 ____ ?_______ 45.8176 14.6600 31.1042 ________ MS _______ ________ 0.4312 0.3747 F P 9.24 0.003 1.15 0.298 Predicted Values for New Observations Obs Fit SE Fit 1 3.2012 0.0706 95% CI 95% PI (3.0614, 3.3410) (1.9594, 4.4431) Unusual Observations Obs 2 9 101 102 105 115 SRES1 2.027999 -4.43288 -2.00435 -2.00137 -0.21259 -2.97678 TRES1 2.055526 -4.83496 -2.03071 -2.02758 -0.21172 -3.08213 HI1 0.056665 0.016012 0.008365 0.027336 0.008554 0.024878 COOK1 0.123525 0.159886 0.016945 0.056286 0.000195 0.113038 DFIT1 0.503787 -0.61677 -0.18651 -0.33991 -0.01967 -0.4923 2 Residual Plots for y Fitted Line Plot Normal Probability Plot y = A+B x 50 89 116 1 52 3 y 32 48 2 119 81 82 71 94 49 103 76 74 99 73 113 100 114 58 20 37 22 111 57 30 5 98 35 28 4 90 66 91 96 64 55 70 11 77 36 13 14 72 105 62 67 46 21 43 31 17 93 63 85 19 61 110 107 99 1 90 0 Residual 2 Percent 4 50 10 -2 0 Residual -3 2 0 30 35 15 10 5 0 -2.25 -1.50 -0.75 0.00 Residual 0.75 1 SRES1 0 -1 -2 50 89 31 91 8 79 96 29 24 118 88 60 75 64 3 117 23 40 55 70 111109 37 95 81 56 17 108 39 25 22 93 112 63 11 34 12 57 7787 82 71 85 30 5 10 53 13 59 49 36 103 26 14 78 729735 98 76 38 99 105 19 41 61 54 16 120 110 1076286 51 33 67 28 90 46 4 21 66 43 113 114 65 58 42 27 20 44 83 92 45 101 6 94 18 74 73 100 47 69 102 52 84 68 104 106 80 15 32 7 119 48 115 -3 -4 9 -5 0.01 0.02 0.03 0.04 3.0 3.2 Fitted Value 3.4 0 -1 -2 2 116 2.8 Versus Order Scatterplot of SRES1 vs HI1 1 2.6 1 Residual Frequency 9 2 115 Histogram 1 25 x -1 1 0.1 115 20 91 31 24 8 118 79 60 88 6 84 75 64 96 17 29 3 117 40 109 23 81 37 2211170 55 56 104 25 95112 68 93 34 39108 11 63 12 106 94 49 71 82 57 87 85 10 53 5 77 80 36 19 13 59 38 76 30 98 14 78 26 97 74 99103 35 72 105 61 41 54 16 110 86120 62 67107 51 33 73 18 7 15 32 100 90 28 46 11366 4 21 43 114 42 65 48119 58 20 44 92 27 69 45 47 102 10183 52 9 20 15 50 89116 1 -2 101 69 102 Versus Fits 2 99.9 79 88 8 118 60 75 29 24 6 84 109 3 117 95 56 23 68 106 40 108 112 104 39 25 12 80 34 87 59 53 38 15 10 26 78 97 18 7 41 54 16 120 33 86 51 65 42 27 44 92 47 83 45 0.05 HI1 2. Does the estimated regression function appear to fit the data well? 0.06 -3 1 10 20 30 40 50 60 70 80 Observation Order 90 100 110 120 3 3. Do outliers and/or leverage points exist? If yes, are the outliers and leverage points valid? 4. What is the point estimate of the change in the mean response when the entrance test score increases by one point? 5. Test whether or not a linear association exists between student’s ACT score (X) and GPA at the end of the freshman year (Y). Use a level of significance of 0.01. (State the alternatives, decision rule, and conclusion.) 6. Obtain a 99% confidence interval for β1. Interpret your confidence interval. 4 7. Find the 90% joint confidence intervals for β0 and β1 and interpret. 8. Obtain a 95% interval estimate of the mean freshman GPA for students whose ACT test score is 28. Interpret your confidence interval. Interval: Interpretation: 9. Ahmad obtained a score of 28 on the entrance test. Predict his freshman GPA using a 95% prediction interval. Interpret your prediction interval. Interval: Interpretation: 10. Compute R-sq and interpret its value? What can we conclude about the model? 5 11. Lack of fit test outputs provided and assume the model assumptions are satisfied: a. Can you perform a lack of fit test for this data set? Explain b. Assume yes, perform the test and write your conclusion. ̅, 𝒀 ̅ ). 12. Verify that your fitted regression line goes through the point (𝑿 6 Part 2: Consider the data sets in Blackboard. Use Minitab to solve the parts below. Save all Minitab output used in a word file called “yourLastName-ID”, and email it to me in blackboard or to mailto:alsawies@kfupm.edu.sabefore the end of the exam session. Question Two: (Production time318) In a manufacturing study, the production times for 111 recent production runs were obtained. The table below lists for each run the production time in hours (7) and the production lot size (X). 1. Based on the scatter plot of the data: Does a linear relation appear adequate here? Would a transformation on X or Y be more appropriate here? Why? 2. obtain the estimated linear regression function and a. Comment on the assumptions. a. Do outliers and/or leverage points exist? If yes, are the outliers and leverage points valid? 3. Use the transformation X' = √𝐗 a. Repeat part 2. b. Estimate y when X=4 7 Question Three: (Sales growth317) A marketing researcher studied annual sales of a product that had been introduced 10 years ago. The data are as follows, where X is the year (coded) and Y is sales in thousands 2 Prepare a scatter plot of the data. Does a linear relation appear adequate here? 3 obtain the estimated linear regression function and a. Comment on the assumptions. b. Do outliers and/or leverage points exist? If yes, are the outliers and leverage points valid? 4 Researcher A suggests the transformation X' = 𝐗 𝟐 and researcher B suggests the transformation Y' = √𝐘 . Which one will you choose and why?