Multiple Regression using STAT 301 Spring 2004 Final Exam Grades
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1 Frequency Frequency Frequency Multiple Regression using STAT 3 Spring 24 Exam Grades We want to figure out the best way to predict a person s final exam (not total course) grade. Possible variables to include are,, Project, Exam and Exam 2. Here is the summary of the individual variables. Do you see any repeated outlier students? Mean = Std. Dev. =.986 = Mean = Std. Dev. = = Mean = Std. Dev. = =
2 Frequency Frequency Frequency Exam Mean = Std. Dev. =.2 = 64 5 Exam Mean = 73.9 Std. Dev. = = Mean = Std. Dev. = 5.2 =
3 ow let s look at the scatterplots between final and each of the other 5 variables: Exam
4 ow let s look at the correlation between the 6 variables: Exam **. Correlation is significant at the. level (2-tailed). *. Correlation is significant at the.5 level (2-tailed). Correlations Exam.744**.387**.322*.57**.67** **.385**.574**.592**.8** **.385**.282*.268*.353** *.574**.282*.553**.66** **.592**.268*.553**.75** **.8**.353**.66**.75** From strongest to weakest, write the order that the other 5 variables are correlated to. Explanatory variable Correlation Which other 2 explanatory variables are most strongly correlated to each other? Which other 2 explanatory variables are the most weakly correlated to each other? 4
5 Let s do regression using all 5 of the variables to explain. The results are: Summary b Adjusted Std. Error of R R Square R Square the Estimate.84 a a. Predictors: (Constant),,, Exam,, b. Dependent Variable: Regression Residual Total AOVA b Sum of Squares df Mean Square F Sig a a. Predictors: (Constant),,, Exam,, b. Dependent Variable: (Constant) Exam a. Dependent Variable: Unstandardized Coe fficients a Standardized 95% Confidence Interval for B t Sig. Lower Bound Upper Bound B Std. Error Beta What is the value for R 2? What is the estimate for the standard deviation? What is the value for the AOVA F-test statistic and its P-value? Are there any individual coefficients which are not significant? If so, which ones? We delete variables one-at-a-time from the model, starting with the least significant. Which one should we delete first? 5
6 Let s do regression using the 4 more significant variables (leaving off Project) to explain. The results are: Summary b Adjusted Std. Error of R R Square R Square the Estimate.83 a a. Predictors: (Constant),,, Exam, b. Dependent Variable: Regression Residual Total AOVA b Sum of Squares df Mean Square F Sig a a. Predictors: (Constant),,, Exam, b. Dependent Variable: (Constant) Exam a. Dependent Variable: Unstandardized Coe fficients a Standardized 95% Confidence Interval for B t Sig. Lower Bound Upper Bound B Std. Error Beta What is the value for R 2? How did it change from the original? Good or bad? What is the estimate for the standard deviation? How did it change from the original? Good or bad? What is the value for the AOVA F-test statistic and its P-value? How did it change from the original? Good or bad? Are there any individual coefficients which are not significant? If so, which ones? We delete variables one-at-a-time from the model, starting with the least significant. Which one should we delete now? 6
7 Let s do regression using the 3 more significant variables (leaving off Project and now Homework) to explain. The results are: Summary b Adjusted Std. Error of R R Square R Square the Estimate.8 a a. Predictors: (Constant),,, Exam b. Dependent Variable: Regression Residual Total AOVA b Sum of Squares df Mean Square F Sig a a. Predictors: (Constant),,, Exam b. Dependent Variable: (Constant) Exam a. Dependent Variable: Unstandardized Coe fficients a Standardized B Std. Error Beta What is the value for R 2? How did it change from the model on the previous page? Good or bad? What is the estimate for the standard deviation? How did it change from the model on the previous page? Good or bad? What is the value for the AOVA F-test statistic and its P-value? How did it change from the model on the previous page? Good or bad? Are there any individual coefficients which are not significant? If so, which ones? We delete variables one-at-a-time from the model, starting with the least significant. Are there any which still need deleting? What is our multiple regression model for Exam? 95% Confidence Interval for B t Sig. Lower Bound Upper Bound Using this model, estimate your Exam score this semester. 7
8 What if we had made a mistake and had deleted Exam and Exam 2 from the model instead of Project and Lab? Our regression output would look like: Summary b Adjusted Std. Error of R R Square R Square the Estimate.662 a a. Predictors: (Constant),,, b. Dependent Variable: Regression Residual Total AOVA b Sum of Squares df Mean Square F Sig a a. Predictors: (Constant),,, b. Dependent Variable: (Constant) a. Dependent Variable: Unstandardized Coe fficients a Standardized 95% Confidence Interval for B t Sig. Lower Bound Upper Bound B Std. Error Beta What is the value for R 2? How did it change from the original? Good or bad? What is the estimate for the standard deviation? How it change from the original? Good or bad? What is the value for the AOVA F-test statistic and its P-value? How did it change from the original? Good or bad? Are there any individual coefficients which are not significant? If so, which ones? How does this model compare to the one you picked on the previous page? Justify your reasons for preferring one over the other. 8
9 General Questions about Multiple Regression State the hypotheses for the original regression AOVA model here. Why do we look at that first? What does R 2 tell us? Answer in terms of this story. What is a good R 2? Where do you find the estimate for the standard deviation? What is a good standard deviation? What do the test statistics and the P-values from the regression output for individual coefficients (t tests) tell you? Write out the hypotheses from the original model and state your conclusions. Why don t we always keep all the variables in the model? How do we know when we re done finding the best model for our data? Is it just when we have only significant coefficients, or is there more than just P-values that we have to consider? What 2 types of graphs (other than scatterplots) should we do to check our assumptions? 9
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