Exam 16: Multiple Regression and Correlation

States Concern over the number of car thefts grew into a project to determine the relationship between car thefts by state and these variables: x₁ = Police per 10,000 persons,by state x₂ = Expenditure by local government for police protection,in thousands,by state x₃ = New passenger car registrations,in thousands,by state. Data from 13 states were collected.The MINITAB regression results are: The regression equation is car-thf $= - 25.3 + 1.28$ police $+ 0.0188$ polexp $+ 0.0969$ registr Predictor Coef Stdev t-ratio p Constant -25.29 17.85 -1.42 0.190 police 1.2831 0.9275 1.38 0.200 polexp 0.018827 0.008460 2.23 0.053 registr 0.09686 0.03536 2.74 0.023 $s = ? ? \quad \mathrm { R } - s q = ? ? \% \quad \mathrm { R } - s q ( a d j ) = ? ? \%$ Analysis of Variance SOURCE DF SS MS F p Regression 3 33007 11002 107.14 0.000 Error 9 924 103 Total 12 33932 Correlation between the variables: car-thf police polexp registr car-thf 1.000 police 0.466 1.000 polexp 0.970 0.390 1.000 registr 0.976 0.406 0.958 1.000 -Perform a test for each partial regression coefficient using a 0.05 significance level.Results: Conclusion: _________________________________________________________

What role does the normal probability plot play in examining whether the multiple regression model is appropriate for a given set of data?

In a multiple regression analysis involving k independent variables and n data points,the degrees of freedom associated with the error sum of squares is:

States Concern over the number of car thefts grew into a project to determine the relationship between car thefts by state and these variables: x₁ = Police per 10,000 persons,by state x₂ = Expenditure by local government for police protection,in thousands,by state x₃ = New passenger car registrations,in thousands,by state. Data from 13 states were collected.The MINITAB regression results are: The regression equation is car-thf $= - 25.3 + 1.28$ police $+ 0.0188$ polexp $+ 0.0969$ registr Predictor Coef Stdev t-ratio p Constant -25.29 17.85 -1.42 0.190 police 1.2831 0.9275 1.38 0.200 polexp 0.018827 0.008460 2.23 0.053 registr 0.09686 0.03536 2.74 0.023 $s = ? ? \quad \text { R-sq } = ? ? \% \quad \text { R-sq(adj) } = ? ? \%$ Analysis of Variance SOURCE DF SS MS F p Regression 3 33007 11002 107.14 0.000 Error 9 924 103 Total 12 33932 Correlation between the variables: car-thf police polexp registr car-thf 1.000 police 0.466 1.000 polexp 0.970 0.390 1.000 registr 0.976 0.406 0.958 1.000 -Compute the multiple standard error of estimate (s_e)from the regression results.

What kind of test is performed in evaluating the overall significance of a multiple regression equation? In evaluating the significance of the individual partial regression coefficients?

What assumptions are required in using the multiple regression model?

The ____________________ is the proportion of the variation in y that is explained by the multiple regression equation.

Motor Vehicle In order to predict motor vehicle purchases for the U.S.,the coefficients of a multiple regression equation were estimated using 25 years of data.The variables were: y = motor vehicle purchases (billions of dollars) x₁ = disposable personal income (billions of dollars) x₂ = U.S.population (millions of persons) x₃ = automobile installment credit (billions of dollars) Part of the results using MINITAB was: The regression equation is $Y = - 61.3 - 0.0022 \times 1 + 0.368 \times 2 + 0.725 \times 3$ Predictor Coef Stdev t-ratio Constant -61.28 36.96 -1.66 X1 -0.00221 0.01504 -0.15 X2 0.3679 0.2001 1.84 X3 0.7254 0.2115 3.43 Analysis of Variance SOURCE DF SS Regression 3 32624 Error 21 591 Total 24 33215 -Use the values in the analysis of variance table to compute the multiple standard error of the estimate.

Salary Data was collected from 40 employees to develop a regression model to predict the employee's annual salary using their years with the company (Years),their starting salary (Starting),and their Gender (Male = 0,Female = 1).The results from Excel regression analysis are shown below: RegresionSlalinlics Multiple R 0.719714957 R Square 0.516551199 Adjusted R Square 0.476253780 Standard Errar 10515.63461 Dbservations 40 $\text { ANOVA }$ of SS MS F Significance F Regression 3 4334682510 1444894170 12.82165585 7.48476-06 Residual 36 4056901131 112691698.1 Total 39 8391583641 Coefficients Standard Error t Stat P -value Intercept 27946.57894 4832.438706 5.783121245 1.35464-06 Years 1665.251558 425.0829092 3.917474737 0.000383313 Starting 0.266374185 0.12610443 2.112330112 0.041661598 Gender -3285.541043 5617.145392 -0.584912943 0.56225464 -In testing the significance of the partial regression coefficient associated with the Starting variable at the 0.05 significance level,what is the appropriate conclusion?

Nutritionist A nutritionist is analyzing the cost of an 8 oz.serving of pasta.The nutritionist anticipates that cost is related to: x₁ = Grams of protein/8 oz. x₂ = Grams of carbohydrates/8 oz. x₃ = Grams of fat/8 oz. Using MINITAB,the nutritionist obtained the following results: The regression equation is $Y = 1.39 + 0.0178 \times 1 - 0.0258 \times 2 - 0.00050 \times 3$ Predictor Coef Stdev t-ratio Constant 1.3928 0.1096 12.71 X1 0.017806 0.006600 2.70 X2 -0.025825 0.001613 -16.01 X3 -0.000501 0.002779 -0.18 $s=0.04805 \quad \mathrm{R}-\mathrm{sq}=97.5 \% \quad \mathrm{R}-\mathrm{sq}(\mathrm{adj})=96.6 \%$ Analysis of Variance SOURCE DF SS MS Regression 3 0.72562 0.24187 Error 8 0.01847 0.00231 Total 11 0.74409 -Test the significance of the regression equation at $\alpha$ = 0.01. Test statistic = ____________________ Critical Value = ____________________ Conclusion: ____________________

In a multiple regression analysis involving 40 observations and 4 independent variables,SST = 375 and SSE = 75.The multiple coefficient of determination is:

In reference to the equation: $\hat { y }$ = -0.25 + 0.08x₁ + 0.10x₂,the value 0.01 is the:

Discuss residual analysis in terms of the assumptions required for using the multiple regression model.

For each y term in the multiple regression equation,the corresponding $\beta$ is referred to as the partial regression coefficient.

Marketing Analyst A marketing analyst is interested in predicting prospective buyer's knowledge about compact disc players.A random sample of 36 buyers was taken,a questionnaire about compact disc players completed,and information about education,income and age was obtained.In estimating the equation,the variables were: y = knowledge about compact disc players x₁ = education (years) x₂ = age x₃ = income (thousands of dollars) The resulting output using MINITAB was: The regression equation is Y=50.2+4.36\times1-0.632\times2-0.068\times3 Predictor Coef Stdev t-ratio Constant 50.168 4.977 10.08 X1 4.3609 0.4064 10.73 X2 -0.63169 0.08172 -7.73 X3 -0.0682 0.1176 -0.58 s=4.615 R-sq =85.0\% R-sq(adj) =83.6\% -Predict the questionnaire score for a buyer who is 41 years of age,has 13 years of education,and $39,000 income.

In a regression model involving 50 observations,the following estimated regression model was obtained. $\hat { Y }$ = 51.4 + 0.70x₁ + 0.679x₂ - 0.378x₃.For this model SST = 120,524 and SSR = 85,400.Then,the value of MSE is:

Professor A statistics professor investigated some of the factors that affect an individual student's final grade in his course.He proposed the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \epsilon$ where: y = final mark (out of 100) x₁ = number of lectures skipped x₂ = number of late assignments x₃ = mid-term test mark (out of 100) The professor recorded the data for 50 randomly selected students.The computer output is shown below. The regression equation is: $\hat { y } = 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ Predictor Coef StDev 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 $S = 13.74$ $\mathrm { R } - \mathrm { Sq } = 30.0 \%$ Analysis of Variance Source of Variation df SS MS F Regression 3 3716 1238.667 6.558 Error 46 8688 188.870 Total 49 12404 -What is the coefficient of determination? What does this statistic tell you?

The computer output for the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \epsilon$ is shown below.However,because of a printer malfunction some of the results are not shown.These are indicated by the boldface letters a to i.Fill in the missing results (up to three decimal places). Predictor Coef StDev T Constant 6.15 4.11 3.51 1.25 -0.71 0.30 $\mathrm { S } = \boldsymbol { d } \quad \mathrm { R } - \mathrm { Sq } = \boldsymbol { e }$ Analysis of Variance Source of Variation df SS MS F Regression 2 412 Error 37 Total 39 974 a = ____________________ b = ____________________ c = ____________________ d = ____________________ e = ____________________ f = ____________________ g = ____________________ h = ____________________ i = ____________________

Salary Data was collected from 40 employees to develop a regression model to predict the employee's annual salary using their years with the company (Years),their starting salary (Starting),and their Gender (Male = 0,Female = 1).The results from Excel regression analysis are shown below: RegresionSlalinlics Multiple R 0.719714957 R Square 0.516551199 Adjusted R Square 0.476253780 Standard Errar 10515.63461 Dbservations 40 $\text { ANOVA }$ of SS MS F Significance F Regression 3 4334682510 1444894170 12.82165585 7.48476-06 Residual 36 4056901131 112691698.1 Total 39 8391583641 Coefficients Standard Error t Stat P -value Intercept 27946.57894 4832.438706 5.783121245 1.35464-06 Years 1665.251558 425.0829092 3.917474737 0.000383313 Starting 0.266374185 0.12610443 2.112330112 0.041661598 Gender -3285.541043 5617.145392 -0.584912943 0.56225464 -In testing the significance of the partial regression coefficient associated with the Years variable at the 0.05 significance level,what is the appropriate conclusion?

Regression Model A multiple regression model was developed to predict the grade point average (GPA)for MBA students based on two entrance exam scores,verbal (VGMAT)and math (MGMAT).The following table shows the actual GPA and predicted GPA for 7 students. Student Actural CPA Predicted CPA 1 3.5 3.71 2 3.1 3.19 3 3.2 3.10 4 4.0 3.74 5 3.6 3.52 6 3.2 3.27 7 3.7 3.78 -Calculate the residual sum of squares.