Exam 17: Multiple Regression

Senior Medical Students A professor of Anatomy wanted to develop a multiple regression model to predict the students' grades in her fourth-year medical course.She decides that the two most important factors are the student's grade point average in the first three years and the student's major.She proposes the model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₃ + $\varepsilon$ ,where y = Fourth-year medical course final score (out of 100),x₁ = G.P.A.in first three years (range from 0 to 12),x₂ = 1 if student's major is medicine and 0 if not,and x₃ = 1 if student's major is biology and 0 if not.The computer output is shown below. THE REGRESSION EQUATION IS y = 9.14 + 6.73x₁ + 10.42x₂ + 5.16x₃ Predictor Coef StDev T Constant 9.14 7.10 1.287 6.73 1.91 3.524 10.42 4.16 2.505 5.16 3.93 1.313 $S = 15.0 \quad R - S q = 44.2 \%$ ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 17098 5699.333 25.386 Error 96 21553 224.510 Total 99 38651 -Most statistical software print a second R² statistic,called the coefficient of determination adjusted for degrees of freedom,which has been adjusted to take into account the sample size and the number of independent variables.

The adjusted coefficient of determination is adjusted for the:

For a multiple regression model,the following statistics are given: Total variation in y = 500,SSE = 80,and n = 25.Then,the coefficient of determination is:

We test an individual coefficient in a multiple regression model using a(n)_________ test.

If the Durbin-Watson statistic,d,has values greater than 2,this indicates

Student's Final Grade A statistics professor investigated some of the factors that affect an individual student's final grade in her course.She proposed the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + €$ ,where y is the final grade (out of 100 points),x₁ is the number of lectures skipped,x₂ is the number of late assignments,and x₃ is the midterm exam score (out of 100).The professor recorded the data for 50 randomly selected students.The computer output is shown below. THE REGRESSION EQUATION IS $\tilde { y } = 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ Predicter Coef StDsv T Constant 41.6 17.8 2.337 -3.18 1.66 -1.916 -1.17 1.13 -1.035 0.63 0.13 4.846 $S = 13.74 \quad R - S q = 30.0 \%$ ANALYSIS OF VARIANCE Source of Variation Repressian 3 3716 1238.667 6.558 Esrar 46 8688 188.870 Total 49 12404 -{Student's Final Grade Narrative} Does this data provide enough evidence to conclude at the 5% significance level that the final grade and the number of skipped lectures are linearly related?

Some of the requirements for the error variable in a multiple regression model are that the standard deviation is a(n)____________________ and the errors are ____________________.

In regression analysis,the total variation in the dependent variable y,measured by $\sum \left( y _ { i } - \bar { y } \right) ^ { 2 }$ ,can be decomposed into two parts: the explained variation,measured by SSR,and the unexplained variation,measured by SSE.

In testing the validity of a multiple regression model in which there are four independent variables,the null hypothesis is:

Consider the following statistics of a multiple regression model: n = 25,k = 5,b₁ = -6.31,and s _{$\varepsilon$} = 2.98.Can we conclude at the 1% significance level that x₁ and y are linearly related?

Real Estate Builder A real estate builder wishes to determine how house size is influenced by family income,family size,and education of the head of household.House size is measured in hundreds of square feet,income is measured in thousands of dollars,and education is measured in years.A partial computer output is shown below. SUMMARY OUTPUT Regression Statistics Multiple R 0.865 R Square 0.748 Adjusted R Square 0.726 Standard Error 5.195 Observations 50 ANOVA Coeff St. Error -value Intercept -1.6335 5.807 -0.281 0.7798 Family Incame 0.4485 0.1137 3.9545 0.0003 Family Size 4.2615 0.8062 5.286 0.0001 Education -0.6517 0.4319 -1.509 0.1383 -{Real Estate Builder Narrative} What percentage of the variability in house size is explained by this model?

A multiple regression is called "multiple" because it has several explanatory variables.

The Durbin-Watson statistic,d,is defined as $d = \sum _ { i = 2 } ^ { n } \left( e _ { i } - e _ { i - 1 } \right) ^ { 2 } / \sum _ { i = 1 } ^ { n } e _ { i }$ ,where e_i is the residual at time period i.

Student's Final Grade A statistics professor investigated some of the factors that affect an individual student's final grade in her course.She proposed the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + €$ ,where y is the final grade (out of 100 points),x₁ is the number of lectures skipped,x₂ is the number of late assignments,and x₃ is the midterm exam score (out of 100).The professor recorded the data for 50 randomly selected students.The computer output is shown below. THE REGRESSION EQUATION IS $\tilde { y } = 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ Predicter Coef StDsv T Constant 41.6 17.8 2.337 -3.18 1.66 -1.916 -1.17 1.13 -1.035 0.63 0.13 4.846 $S = 13.74 \quad R - S q = 30.0 \%$ ANALYSIS OF VARIANCE Source of Variation Repressian 3 3716 1238.667 6.558 Esrar 46 8688 188.870 Total 49 12404 -{Student's Final Grade Narrative} What is the adjusted coefficient of determination? What does this statistic tell you?

Real Estate Builder A real estate builder wishes to determine how house size is influenced by family income,family size,and education of the head of household.House size is measured in hundreds of square feet,income is measured in thousands of dollars,and education is measured in years.A partial computer output is shown below. SUMMARY OUTPUT Regression Statistics Multiple R 0.865 R Square 0.748 Adjusted R Square 0.726 Standard Error 5.195 Observations 50 ANOVA Coeff St. Error -value Intercept -1.6335 5.807 -0.281 0.7798 Family Incame 0.4485 0.1137 3.9545 0.0003 Family Size 4.2615 0.8062 5.286 0.0001 Education -0.6517 0.4319 -1.509 0.1383 -{Real Estate Builder Narrative} At the 0.01 level of significance,what conclusion should the builder draw regarding the inclusion of education in the regression model?

Student's Final Grade A statistics professor investigated some of the factors that affect an individual student's final grade in her course.She proposed the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + €$ ,where y is the final grade (out of 100 points),x₁ is the number of lectures skipped,x₂ is the number of late assignments,and x₃ is the midterm exam score (out of 100).The professor recorded the data for 50 randomly selected students.The computer output is shown below. THE REGRESSION EQUATION IS $\tilde { y } = 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ Predicter Coef StDsv T Constant 41.6 17.8 2.337 -3.18 1.66 -1.916 -1.17 1.13 -1.035 0.63 0.13 4.846 $S = 13.74 \quad R - S q = 30.0 \%$ ANALYSIS OF VARIANCE Source of Variation Repressian 3 3716 1238.667 6.558 Esrar 46 8688 188.870 Total 49 12404 -{Student's Final Grade Narrative} Does this data provide enough evidence at the 1% significance level to conclude that the final grade and the midterm exam score are positively linearly related?