Exam 13: Multiple Regression

Exhibit 13-2 A regression model between sales Y in $1,000), unit price X1 in dollars) and television advertisement X2 in dollars) resulted in the following function: $\hat { Y }$ =7-3X1+5X2 For this model SSR = 3500, SSE = 1500, and the sample size is 18. -Refer to Exhibit 13-2. To test for the significance of the model, the p-value is

The following regression model has been proposed to predict sales at a computer store. $\hat { Y }$ = 50 - 3X1 + 20X2 + 10X3 where X1 = competitor's previous days sales in $1,000s) X2 = population within 1 mile in 1,000s) X3 = = sales in $1,000s) $\hat { Y }$ Predict sales in dollars) for a store with the competitor's previous day's sale of $5,000, a population of 20,000 within 1 mile, and nine radio advertisements.

Below you are given a partial computer output based on a sample of 12 observations relating the number of personal computers sold by a computer shop per month Y), unit price X1 in $1,000) and the number of advertising spots X2) they used on a local television station. ANOVA DF SS MS F Significance F Regression 2 655.955 Residual 9 Total 838.917 a. At α = 0.05 level of significance, test to determine if the model is significant. That is, determine if there exists a significant relationship between the independent variables and the dependent variable. b. Determine the multiple coefficient of determination. c. Determine the adjusted multiple coefficient of determination.

Exhibit 13-4 a. Y=β₀ + β₁X1 + β2X2 + ε b. EY)=β₀ + β₁X1 + β2X2 + ε c. $\hat { Y }$ =b₀ + b₁X1 + b2X2 d. EY)=β₀ + β₁X1 + β2X2 -Refer to Exhibit 13-4. Which equation describes the multiple regression model?

Exhibit 13-8 The following estimated regression model was developed relating yearly income Y in $1,000s) of 30 individuals with their age X1) and their gender X2) 0 if male and 1 if female). $\hat { Y }$ =30+0.7X1+3X2 Also provided are SST = 1,200 and SSE = 384. -Refer to Exhibit 13-8. The yearly income of a 24-year-old female individual is

Exhibit 13-6 Below you are given a partial computer output based on a sample of 16 observations. Coefficient Standard Error Intercept 12.924 4.425 -3.682 2.63. 45.216 12.560 $\text { Analysis of Variance }$ Source of Degrees Sum of Mean Variation of Freedom Squares Square F Regression 4,853 2,426.5 Error 485.3 -Refer to Exhibit 13-6. The t value obtained from the table which is used to test an individual parameter at the 1% level is

The following results were obtained from a multiple regression analysis of supermarket profitability. The dependent variable, Y, is the profit in thousands of dollars) and the independent variables, X1 and X2, are the food sales and nonfood sales also in thousands of dollars). ANOVA DF SS Regression 2 562.363 Residual 9 225.326 Total Coefilicient Standard Error Intercept -15.0621 0.0972 0.054 0.2484 0.092 a. Write the estimated regression equation for the relationship between the variables. b. Compute the coefficient of determination and fully interpret its meaning. c. Carry out a test of whether Y is significantly related to the independent variables. Use a 0.05 level of significance. d. Carry out a test to determine if there is a significant relationship between X1 and Y. Use a .05 level of significance. e. How many supermarkets were in the sample used here?

Exhibit 13-5 Below you are given a partial Minitab output based on a sample of 25 observations. Coefficient Standard Error Constant 145.321 48.682 25.625 9.150 -5.720 3.575 0.823 0.183 -Refer to Exhibit 13-5. Carry out the test of significance for the parameter ?? at the 5% level. The null hypothesis should be

Exhibit 13-9 In a regression analysis involving 25 observations, the following estimated regression equation was developed. $\hat { Y }$ =10 - 18X1+3X2+14X3 Also, the following standard errors and the sum of squares were obtained. Sb₁ = 3 Sb2 = 6 Sb3 = 7 SST = 4,800 SSE = 1,296 -Refer to Exhibit 13-9. If we are interested in testing for the significance of the relationship among the variables i.e., significance of the model) the critical value of F at ? = 0.05 is

Exhibit 13-12 In a laboratory experiment, data were gathered on the life span Y in months) of 33 rats, units of daily protein intake X1), and whether or not agent X2 a proposed life extending agent) was added to the rats diet X2 = 0 if agent X2 was not added, and X2 = 1 if agent was added.) From the results of the experiment, the following regression model was developed. $\hat { Y }$ =36+0.8X1 - 1.7X2 Also provided are SSR = 60 and SST = 180. -Refer to Exhibit 13-12. If we want to test for the significance of the model, the critical value of F at 95% confidence is

Exhibit 13-10 In a regression model involving 30 observations, the following estimated regression equation was obtained. $\hat { Y }$ =170+34X1 - 3X2+8X3+58X4+3X5 For this model, SSR = 1,740 and SST = 2,000. -Refer to Exhibit 13-10. The degrees of freedom associated with SST are

The following results were obtained from a multiple regression analysis. Source of Degrees Sum of Mean Variation of Freedom Squares Square F Regression 384 48 Error 20 Total 704 a. How many independent variables were involved in this model? b. How many observations were involved? c. Determine the F statistic.

Exhibit 13-9 In a regression analysis involving 25 observations, the following estimated regression equation was developed. $\hat { Y }$ =10 - 18X1+3X2+14X3 Also, the following standard errors and the sum of squares were obtained. Sb₁ = 3 Sb2 = 6 Sb3 = 7 SST = 4,800 SSE = 1,296 -Refer to Exhibit 13-9. The coefficient of X3

In order to test for the significance of a regression model involving 3 independent variables and 47 observations, the numerator and denominator degrees of freedom respectively) for the critical value of F are

The correct relationship between SST, SSR, and SSE is given by

Below you are given a partial computer output relating the price of a company's stock Y in dollars), the Dow Jones industrial average X1), and the stock price of the company's major competitor X2 in dollars). ANOVA Regression DF SS MS F Residual 20 40 Total 800 a. What has been the sample size for this regression analysis? b. At α = 0.05 level of significance, test to determine if the model is significant. That is, determine if there exists a significant relationship between the independent variables and the dependent variable. c. Determine the multiple coefficient of determination.

Exhibit 13-12 In a laboratory experiment, data were gathered on the life span Y in months) of 33 rats, units of daily protein intake X1), and whether or not agent X2 a proposed life extending agent) was added to the rats diet X2 = 0 if agent X2 was not added, and X2 = 1 if agent was added.) From the results of the experiment, the following regression model was developed. $\hat { Y }$ =36+0.8X1 - 1.7X2 Also provided are SSR = 60 and SST = 180. -Refer to Exhibit 13-12. The life expectancy of a rat that was given 3 units of protein daily, and who took agent X2 is

In a regression model involving 46 observations, the following estimated regression equation was obtained. $\hat { Y }$ =17+4X1 - 3X2+8X3+5X4+8X5 For this model, SST = 3410 and SSE = 510. a. Compute the coefficient of determination. b. Perform an F test and determine whether or not the regression model is significant.

A variable that cannot be measured in numerical terms is called

A regression model relating a dependent variable, Y, with one independent variable, X1, resulted in an SSE of 400. Another regression model with the same dependent variable, Y, and two independent variables, X1 and X2, resulted in an SSE of 320. At α = .05, determine if X2 contributed significantly to the model. The sample size for both models was 20.