Exam 13: Multiple Regression

In a regression model involving more than one independent variable, which of the following tests must be used in order to determine if the relationship between the dependent variable and the set of independent variables is significant?

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 multiple coefficient of determination is

The following regression model has been proposed to predict monthly sales at a shoe store. $\hat { Y }$ = 40 - 3X1 + 12X2 + 10X3 where X1 = competitor's previous month's sales in $1,000s) X2 = store's previous month's sales in $1,000s) X3 = = sales in $1,000s) $\hat { Y }$ a. Predict sales in dollars) for the shoe store if the competitor's previous month's sales were $9,000, the store's previous month's sales were $30,000, and no radio advertisements were run. b. Predict sales in dollars) for the shoe store if the competitor's previous month's sales were $9,000, the store's previous month's sales were $30,000, and 10 radio advertisements were run.

Exhibit 13-7 A regression model involving 4 independent variables and a sample of 15 periods resulted in the following sum of squares. SSR = 165 SSE = 60 -Refer to Exhibit 13-7. The coefficient of determination is

Exhibit 13-1 In a regression model involving 44 observations, the following estimated regression equation was obtained. $\hat { Y }$ = 29+18X1+43X2+87X3 For this model SSR = 600 and SSE = 400. -Refer to Exhibit 13-1. MSR for this model is

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. The interpretation of the coefficient on X1 is that

A variable that takes on the values of 0 or 1 and is used to incorporate the effect of qualitative variables in a regression model is called

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 p-value for testing the significance of the regression model 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 test statistic F for testing the significance of the above model is

In order to test for the significance of a regression model involving 4 independent variables and 36 observations, the numerator and denominator degrees of freedom respectively) for the critical value of F are

Exhibit 13-7 A regression model involving 4 independent variables and a sample of 15 periods resulted in the following sum of squares. SSR = 165 SSE = 60 -Refer to Exhibit 13-7. The test statistic from the information provided is

A multiple regression model has the form Y = 70 - 14X1 + 5X2 As X1decreases by 1 unit holding X2 constant), Y is expected to

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 not given any protein and that did not take agent X2 is

Exhibit 13-3 In a regression model involving 30 observations, the following estimated regression equation was obtained: $\hat { Y }$ =17+4X1 - 3X2+8X3+8X4 For this model SSR = 700 and SSE = 100. -Refer to Exhibit 13-3. The computed F statistic for testing the significance of the above model is

A regression model involving 8 independent variables for a sample of 69 periods resulted in the following sum of squares. SSE = 306 SST = 1800 a. Compute the coefficient of determination. b. At α = 0.05, test to determine whether or not the model is significant.

A student used multiple regression analysis to study how family spending Y) is influenced by income X1), family size X2), and additions to savings X3). The variables Y, X1, and X3 are measured in thousands of dollars. The following results were obtained. ANOVA DF SS Regression 3 45.9634 Residual 11 2.6218 Total Coefilicient Standard Error Intercept 0.0136 0.7992 0.074 0.2280 0.190 -0.5796 0.920 a. Write out the estimated regression equation for the relationship between the variables. b. Compute R². What can you say about the strength of this relationship? c. Carry out a test of whether Y is significantly related to the independent variables. Use a .05 level of significance. d. Carry out a test to see if X3 and Y are significantly related. Use a .05 level of significance.

Sherri Cola Company has developed a regression model relating its sales Y in $10,000s) with four independent variables. The four independent variables are price per unit PRICE, in dollars), competitor's price COMPRICE, in dollars), advertising ADV, in $1,000s) and type of container used CONTAIN; 1 = Cans and 0 = Bottles). Part of the regression results is shown below. Assume n = 25) Coefilicient Standard Error Intercept 443.143 PRICE -57.170 20.426 COMPRICE 27.681 19.991 ADV 0.025 0.023 CONTAIN -95.353 91.027 a. If the manufacturer uses can containers, his price is $1.25, advertising $200,000, and his competitor's price is $1.50, what is your estimate of his sales? Give your answer in dollars. b. Test to see if there is a significant relationship between sales and unit price. Let α = 0.05. c. Test to see if there is a significant relationship between sales and advertising. Let α = 0.05. d. Is the type of container a significant variable? Let α = 0.05. e. Test to see if there is a significant relationship between sales and competitor's price. Let α = 0.05.

The Brock Juice Company has developed a regression model relating sales Y in $10,000s) with four independent variables. The four independent variables are price per unit X1, in dollars), competitor's price X2, in dollars), advertising X3, in $1,000s) and type of container used X4) 1 = Cans and 0 = Bottles). Part of the regression results are shown below: Analysis of Variance Source of Variation Degrees of Freedom Sum of Squares Regression 4 283,940.60 Error Residuals) 18 621,735.14 a. Compute the coefficient of determination and fully interpret its meaning. b. Is the regression model significant? Explain what your answer implies. Let α = 0.05. c. What has been the sample size for this analysis?

In order to test for the significance of a regression model involving 14 independent variables and 255 observations, the numerator and denominator degrees of freedom respectively) for the critical value of F are

Multiple regression analysis was used to study the relationship between a dependent variable, Y, and three independent variables X1, X2 and, X3. The following is a partial result of the regression analysis involving 20 observations. F Coefficient Standard Error Intercept 20.00 5.00 15.00 3.00 8.00 5.00 -18.00 10.00 $\text { Analysis of Variance }$ Source DF SS MS Regression 80 Error 320 Coefficient Standard Error Intercept 20.00 5.00 15.00 3.00 8.00 5.00 -18.00 10.00 $\text { Analysis of Variance }$ Source DF SS MS Regression 80 Error 320 a. Compute the coefficient of determination. b. Perform a t test and determine whether or not β₁ is significantly different from zero α = 0.05). c. Perform a t test and determine whether or not β2 is significantly different from zero α = 0.05). d. Perform a t test and determine whether or not β3 is significantly different from zero α = 0.05). e. At α = 0.05, perform an F test and determine whether or not the regression model is significant.