Week 2 Homework Assignment A college Admissions Officer is interested in determining the extent to which a prospective student's high school GPA and/or SAT score can be used as a basis for predicting his or her college freshman GPA. He believes that prospective students who have higher high school GPAs and SAT scores will have a higher college freshman GPA. He randomly selected forty students who recently completed their freshman year and collected the information reflected in the following table. Student No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 College Freshma n GPA 3.56 2.59 3.09 3.68 2.91 2.48 2.34 2.38 2.58 2.70 2.54 2.89 3.02 3.33 3.16 2.60 2.65 3.27 2.89 2.91 2.85 3.65 3.54 3.83 3.05 3.15 2.35 3.06 3.25 3.49 High School GPA 3.95 2.82 3.29 3.84 2.97 3.09 2.85 2.84 3.00 3.06 2.82 3.14 3.21 3.46 3.23 3.25 3.23 3.89 3.36 3.31 3.16 3.97 3.77 3.99 3.11 3.94 2.86 3.65 3.78 3.97 1 of 3 SAT Score 1383 1014 1217 1458 1157 1238 856 880 959 1011 959 1100 1156 1282 1227 1266 1291 1168 1041 1060 1044 1350 1319 1436 1150 1498 1117 1458 1134 1230 Student No. 31 32 33 34 35 36 37 38 39 40 College Freshma n GPA 3.02 2.91 3.48 3.09 3.66 2.31 2.49 2.42 2.78 2.98 High School GPA 3.36 3.16 3.70 3.22 3.73 2.89 3.04 2.88 3.23 3.38 SAT Score 1075 1043 1258 1128 1343 1069 1154 1122 1292 1015 1. Perform a simple linear regression with a 95% confidence level using college freshman GPA as the dependent variable and high school GPA as the independent variable, and evaluate the statistical significance of the regression model. 2. Perform a simple linear regression with a 95% confidence using college freshman GPA as the dependent variable and SAT score as the independent variable, and evaluate the statistical significance of the regression model. 3. Compare the two simple linear regression models, select your preferred simple regression model, and explain the basis for selecting your preferred model. 4. Perform a multiple linear regression with a 95% confidence using college freshman GPA as the dependent variable and high school GPA and SAT score as the independent variables. a. Evaluate the statistical significance of the regression model as a whole. b. Evaluate the statistical significance of the linear relationship between the dependent variable and each independent variable. c. Discuss the extent to which there is evidence of multicollinearity between the independent variables. 5. Compare your preferred simple linear regression model (i.e., the regression model you selected in step 3) to the multiple linear regression model. Discuss whether the simple regression model or multiple regression model would be your overall preferred regression model, including explaining the basis for selecting your preferred model. 6. Discuss the contribution of each independent variable for your overall preferred regression model (i.e., the model you selected in step 5) to predicting the value of the dependent variable. Round the coefficients to four decimal places. 7. Discuss the range of values for the independent variable(s) for your preferred regression model (i.e., the regression model you selected in step 5) for which the regression model is valid. 8. Discuss the p-value for the coefficient for the y-intercept for your overall preferred regression model, including explaining why a p-value that is not less than or equal to 2 of 3 = 0.05 would not be cause for rejecting the regression model. (Hint: Consider the range of values for the independent variables associated with the given data set.) 9. Identify the regression equation associated with your overall preferred regression model and associated degree of error associated with using the model to predict a student's college freshman GPA. Round the coefficients and degree of error to four decimal places. 10. Calculate the predicted college freshman GPA for a student with a high school GPA of 3.25 and an SAT score of 1115 using your overall preferred regression model. Round your answer to four decimal places. 11. Identify the lower and upper limits associated with a 95% confidence level interval estimate for the predicted college freshman GPA for a student with a high school GPA of 3.25 and an SAT score of 1115 using your overall preferred regression model. Round the coefficients and your final answers to four decimal places. 3 of 3 Student No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 College Freshman GPA 3.56 2.59 3.09 3.68 2.91 2.48 2.34 2.38 2.58 2.7 2.54 2.89 3.02 3.33 3.16 2.6 2.65 3.27 2.89 2.91 2.85 3.65 3.54 3.83 3.05 3.15 2.35 3.06 3.25 3.49 3.02 2.91 3.48 3.09 3.66 2.31 2.49 2.42 2.78 2.98 High School GPA 3.95 2.82 3.29 3.84 2.97 3.09 2.85 2.84 3 3.06 2.82 3.14 3.21 3.46 3.23 3.25 3.23 3.89 3.36 3.31 3.16 3.97 3.77 3.99 3.11 3.94 2.86 3.65 3.78 3.97 3.36 3.16 3.7 3.22 3.73 2.89 3.04 2.88 3.23 3.38 SAT Score 1383 1014 1217 1458 1157 1238 856 880 959 1011 959 1100 1156 1282 1227 1266 1291 1168 1041 1060 1044 1350 1319 1436 1150 1498 1117 1458 1134 1230 1075 1043 1258 1128 1343 1069 1154 1122 1292 1015 Min Max 2.82 3.99 SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.8946952934 0.800479668 0.795229133 0.1917931317 40 ANOVA df Regression Residual Total Intercept High School GPA SS 1 5.6080624956 38 1.3978150044 39 7.0058775 MS 5.6080624956 0.0367846054 Coefficients Standard Error -0.3438942243 0.2703586746 0.9946459443 0.0805554901 t Stat -1.2719925664 12.3473389911 RESIDUAL OUTPUT Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Predicted College Freshman GPA 3.5849572558 2.4610073387 2.9284909325 3.4755462019 2.6102042303 2.7295617436 2.490846717 2.4809002576 2.6400436087 2.6997223653 2.4610073387 2.7792940409 2.848919257 3.097580743 2.8688121758 2.8887050947 2.8688121758 3.5252784991 2.9981161486 2.9483838514 2.7991869597 3.6048501746 3.4059209858 3.6247430935 2.7494546625 Residuals Standard Residuals -0.024957256 -0.1318269743 0.1289926613 0.6813534473 0.1615090675 0.8531086868 0.2044537981 1.0799474848 0.2997957697 1.5835542818 -0.249561744 -1.3182126223 -0.150846717 -0.7967889769 -0.100900258 -0.5329662759 -0.060043609 -0.3171569555 0.0002776347 0.001466497 0.0789926613 0.4172479392 0.1107059591 0.5847610719 0.171080743 0.9036673315 0.232419257 1.2276641192 0.2911878242 1.5380861652 -0.288705095 -1.5249721149 -0.218812176 -1.1557900178 -0.255278499 -1.3484091544 -0.108116149 -0.5710814073 -0.038383851 -0.2027477315 0.0508130403 0.2684000763 0.0451498254 0.2384863513 0.1340790142 0.7082201236 0.2052569065 1.0841895916 0.3005453375 1.5875135814 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 3.5750107963 2.5007931764 3.2865634725 3.4158674452 3.6048501746 2.9981161486 2.7991869597 3.3362957697 2.8588657164 3.366135148 2.5306325548 2.6798294464 2.5206860953 2.8688121758 3.0180090675 -0.425010796 -0.150793176 -0.226563472 -0.165867445 -0.114850175 0.0218838514 0.1108130403 0.1437042303 0.2311342836 0.293864852 -0.220632555 -0.189829446 -0.100686095 -0.088812176 -0.038009067 -2.2449538465 -0.7965061698 -1.1967332204 -0.8761301181 -0.6066512747 0.1155929138 0.5853266861 0.7590615754 1.2208767483 1.5522265212 -1.1654054598 -1.0027000481 -0.5318350474 -0.4691156966 -0.2007680817 F Significance F 152.4568 0.0000 P-value 0.2111 0.0000 Residuals Lower 95% Upper 95% Lower 95.0% Upper 95.0% -0.8912067473 0.203418299 -0.89120675 0.203418299 0.8315698802 1.157722008 0.83156988 1.157722008 SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.6741923603 0.4545353386 0.4401810054 0.3171193781 40 ANOVA df Regression Residual Total SS 3.1844189018 3.8214585982 7.0058775 MS 3.1844189018 0.1005647 Coefficients Standard Error 0.8631634399 0.3783173555 0.0017974246 0.0003194172 t Stat 2.2815856246 5.62719954 1 38 39 Intercept SAT Score RESIDUAL OUTPUT Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Predicted College Freshman GPA 3.3490016039 2.6857519419 3.0506291272 3.4838084458 2.9427836537 3.088375043 2.4017588617 2.4448970511 2.5868935912 2.6803596682 2.5868935912 2.8403304539 2.9409862292 3.1674617235 3.0686033728 3.1387029306 3.1836385445 2.9625553239 2.734282405 2.7684334716 2.7396746787 3.2896865935 3.2339664322 3.4442651055 2.9302016818 3.5557054281 2.8708866714 3.4838084458 Residuals Standard Residuals 0.2109983961 0.6740573958 -0.0957519419 -0.3058900248 0.0393708728 0.1257745483 0.1961915542 0.6267553241 -0.0327836537 -0.1047309585 -0.608375043 -1.943520447 -0.0617588617 -0.1972954215 -0.0648970511 -0.2073207099 -0.0068935912 -0.0220223292 0.0196403318 0.0627431825 -0.0468935912 -0.1498066931 0.0496695461 0.1586747838 0.0790137708 0.2524181111 0.1625382765 0.5192462568 0.0913966272 0.2919764967 -0.5387029306 -1.720945283 -0.5336385445 -1.7047665493 0.3074446761 0.9821655594 0.155717595 0.4974568457 0.1415665284 0.4522497196 0.1103253213 0.3524462753 0.3603134065 1.1510604865 0.3060335678 0.97765762 0.3857348945 1.2322722033 0.1197983182 0.3827087971 -0.4057054281 -1.2960702515 -0.5208866714 -1.6640292993 -0.4238084458 -1.3539023165 29 30 31 32 33 34 35 36 37 38 39 40 2.9014428889 3.0739956465 2.79539484 2.7378772541 3.1243235341 2.8906583416 3.2771046216 2.7846102926 2.9373913801 2.8798737942 3.1854359691 2.6875493665 0.3485571111 0.4160043535 0.22460516 0.1721227459 0.3556764659 0.1993416584 0.3828953784 -0.4746102926 -0.4473913801 -0.4598737942 -0.4054359691 0.2924506335 1.1135037182 1.3289712924 0.7175256876 0.54986489 1.1362472737 0.6368186756 1.2232010594 -1.5161943586 -1.429240573 -1.4691170067 -1.2952094354 0.9342654545 F Significance F 31.66537 0.0000 P-value 0.0282 0.0000 Residuals Lower 95% Upper 95% Lower 95.0% Upper 95.0% 0.0972999933 1.629026887 0.0972999933 1.629026887 0.0011507982 0.002444051 0.0011507982 0.002444051 Both models used the simple linear regression model and have an equal number of independent variables. Therefore, both models are accurately tested using R2 and the Adjusted R2 and can be used. The model in section #1 has a higher R2 and adjusted R2 than section #2. Therefore, section #1 is the preferred model. SUMMARY OUTPUT In this case the p value for the F test is smaller than the signifi conclude that the regression model is significant as a whole. Regression Statistics Multiple R 0.8948105494 R Square 0.8006859194 Adjusted R Square 0.7899121853 Standard Error 0.194267163 Observations 40 The p-value for the t-test is larget than 0.05 for the SAT score linear relationship between SAT and the dependent variable. relationship between the High School GPA and the dependen Due to one of the independent variables being insignificant t on this model. ANOVA df Regression Residual Total Intercept High School GPA SAT Score SS MS F 5.609507467 2.8047537335 74.3183294 1.396370033 0.0377397306 7.0058775 2 37 39 Coefficients -0.3519559052 0.9769325956 5.7187868905987E-005 Standard Error t Stat 0.2769280503 -1.2709290545 0.1218709499 8.0161235821 0.0002922629 0.1956726992 RESIDUAL OUTPUT Observation Predicted College Freshman GPA 1 3.5860186703 2 2.4609825136 3 2.931749971 4 3.482845175 5 2.6157002682 6 2.7375643971 7 2.4812548082 8 2.4728579911 9 2.633685048 10 2.695274773 11 2.4578371808 12 2.7785191009 13 2.8501069033 14 3.1015457237 15 2.8737058939 16 2.8954748727 17 2.8773659175 18 3.5151073228 19 2.9900701877 20 2.9423101274 21 2.7948552322 22 3.6036701226 23 3.4065107795 24 3.6281269312 25 2.7520705165 26 3.5828259493 27 2.5059501679 Residuals Standard Residuals -0.0260186703 -0.137504573 0.1290174864 0.6818370869 0.158250029 0.8363264688 0.197154825 1.0419321855 0.2942997318 1.555327711 -0.2575643971 -1.3611872553 -0.1412548082 -0.7465094044 -0.0928579911 -0.4907398517 -0.053685048 -0.28371702 0.004725227 0.0249720804 0.0821628192 0.4342175533 0.1114808991 0.589158986 0.1698930967 0.8978582468 0.2284542763 1.2073448538 0.2862941061 1.5130192407 -0.2954748727 -1.5615381456 -0.2273659175 -1.201593049 -0.2451073228 -1.2953535803 -0.1000701877 -0.5288551745 -0.0323101274 -0.1707539326 0.0551447678 0.2914314089 0.0463298774 0.2448461023 0.1334892205 0.7054694971 0.2018730688 1.0668673604 0.2979294835 1.5745103768 -0.4328259493 -2.2874169435 -0.1559501679 -0.8241720652 P-value 0.2117 0.0000 0.8459 28 29 30 31 32 33 34 35 36 37 38 39 40 3.2972279818 3.4057003497 3.5968075783 2.9920145753 2.7947980443 3.3346370378 2.8582749689 3.3688059845 2.5325131281 2.6839139863 2.5257747592 2.8774231054 3.008121955 -0.2372279818 -1.2537125048 -0.1557003497 -0.8228518152 -0.1068075783 -0.564461222 0.0279854247 0.1478985602 0.1152019557 0.6088241839 0.1453629622 0.7682203511 0.2317250311 1.2246302774 0.2911940155 1.5389144897 -0.2225131281 -1.1759468216 -0.1939139863 -1.0248048635 -0.1057747592 -0.5590029359 -0.0974231054 -0.5148657614 -0.028121955 -0.1486201013 is smaller than the significance level of 0.05, so we can is significant as a whole. an 0.05 for the SAT score indicating that there is no significant the dependent variable. However, there is a significant linear ol GPA and the dependent variable. ables being insignificant there is no evidence of multicollinearity Significance F 0.0000 Lower 95% -0.9130654335 0.7299985954 -0.000534993 Upper 95% Lower 95.0% Upper 95.0% 0.209153623 -0.91306543 0.209153623 1.223866596 0.7299986 1.223866596 0.000649369 -0.00053499 0.000649369 The Adjusted R2 is the best measure to compare models since the number of independent variables differs between the two models. Based on the adjusted R2, the preferred model would be the simple linear regression model because the adjusted R2 is higher than in the multiple linear regression model. (model in section #1) In the preferred model (in section #1) the only independent variable High School GPA. The coefficient for the independent variable in the preferred model is 0.9946. The independent variable on High School GPA is contributing to the prediction of the dependent variable. The other independent variables i.e. SAT score, College GPA play no role in this prediction. The regression model is valid based on the range of independent variables. The current model in section #1 is based on the range High School GPA 2.82 to 3.99, which makes this regression model valid based on the range of independent variables. The p-value for the Y intercept for the preferred model (in section #1) is 0.2111. This p-value is greater than 0.05 so the Y intercept is not significant. However, a p-value that is not less than or equal to ?? = 0.05 would not be causefor rejecting the regression model because the value for X = 0 and is not observed in the range for which the regression model is based on. Therefore, it is an invalid point which should not be considered. The regression equation associated with the overall preferred regression model is College Freshman GPA = -0.3439 + 0.9946*High School GPA The degree of error associated with using the model to predict a student's college freshman GPA is 0.1917. Confidence Interval Estimate Data X Value Confidence Level 3.25 95% Intermediate Calculations Sample Size Degrees of Freedom t Value XBar, Sample Mean of X Sum of Squared Differences from XBar Standard Error of the Estimate h Statistic Predicted Y (YHat) 40 38 2.0244 3.3855 5.8462 0.1913 0.0281 2.8882 For Individual Response Y Interval Half Width Prediction Interval Lower Limit Prediction Interval Upper Limit 0.3927 2.4955 3.2809 The predicted College Freshman GPA School GPA of 3.25 and a SAT Score o preferred regression model is 2.8888 predicted College Freshman GPA for a student with a High ool GPA of 3.25 and a SAT Score of 1115 using the overall erred regression model is 2.8888. The lower and upper limits associated with the 95% confidence level interval estimate for the predicted college freshman GPA for a student with a high school GPA of 3.25 and a SAT score of 1115 using the overall preferred regression mdel is as follows: - lower limit = 2.4964 - upper limit = 3.2811