Exam 16: Multiple Regression and Correlation

For y = annual household expenditure for appliances maintenance and repair,what independent variables could help explain the amount of money a household spends per year for this purpose?

Under what circumstances would a multiple regression analysis be preferable to a simple regression analysis?

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 -For a male employee with 5 years of experience and a starting salary of $30,000,what is the approximate 95% confidence interval for his annual salary?

What is the coefficient of multiple determination,and what does it tell us about the relationship between y and the independent variables.

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 -Do these data provide enough evidence to conclude at the 5% significance level that the model is useful in predicting the final mark? Test statistic = ____________________ Critical Value = ____________________ Conclusion: ____________________

Grade A statistics teacher collected the following data to determine if the number of hours a student studied during the semester and the number of classes missed could be used to predict the final grade for the course.The following table shows the results of the model being applied to 8 students. Student Predicted Grade Actural Grade 1 84 92 2 93 95 3 77 81 4 81 78 5 81 75 6 89 88 7 88 85 85 85 -Calculate the residual sum of squares.

It is a good idea to make point estimates based on x values that lie beyond the range of the underlying data.

What is a dummy variable,and how is it useful to multiple regression? Give an example of three dummy variables that could be used in describing your home town.

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 total sum of squares.

In order to test the significance of a multiple regression model involving 4 independent variables and 30 observations,the numerator and denominator degrees of freedom (respectively)for the critical value of F are:

In interpreting the multiple regression equation,it can be a mistake to conclude that one independent variable is more important than another just because its partial regression coefficient happens to be large.

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 coefficient of multiple determination.

Among the results of a multiple regression analysis are the following sum-of-squares terms: SST,SSR,and SSE.What does each term represent,and how do the terms contribute to our understanding of the relationship between y and the set of independent variables.

The amount of variation in the dependent variable that is not explained by the multiple regression equation is known as:

In multiple regression,to what does the descriptor "multiple" refer?

A dummy variable is used to incorporate qualitative data into the analysis.

Grade A statistics teacher collected the following data to determine if the number of hours a student studied during the semester and the number of classes missed could be used to predict the final grade for the course.The following table shows the results of the model being applied to 8 students. Student Predicted Grade Actural Grade 1 84 92 2 93 95 3 77 81 4 81 78 5 81 75 6 89 88 7 88 85 85 85 -Calculate the total sum of squares.

Grade A statistics teacher collected the following data to determine if the number of hours a student studied during the semester and the number of classes missed could be used to predict the final grade for the course.The following table shows the results of the model being applied to 8 students. Student Predicted Grade Actural Grade 1 84 92 2 93 95 3 77 81 4 81 78 5 81 75 6 89 88 7 88 85 85 85 -Calculate the multiple standard error of estimate.

Equation The regression equation, $\hat{ Y }$ = 4 + 1.5x₁ + 2.5x₂ has been fitted to 25 data points.The means of x₁ and x₂ are 30 and 46,respectively.The sum of the squared differences between observed and predicted values of y has been calculated as SSE = 175,and the sum of the squared differences between y values and mean of y is SST = 525. -What is the approximate 95% confidence interval for the mean of y whenever x₁ = 20 and x₂ = 25.

A dummy variable will have a value of either ____________________ or ____________________,depending on whether a given characteristic is present or absent.