Exam 15: Multiple Regression

A measure of identifying the effect of an unusual x value on the regression results is called

The following estimated regression equation was developed relating yearly income (y in $1000s) of 30 individuals with their age (x₁) and their gender (x₂) (0 if male and 1 if female). $\hat { y }$ = 30 + .7x₁ + 3x₂ Also provided are SST = 1200 and SSE = 384.The yearly income (in $) expected of a 24-year-old male individual is

The following estimated regression equation has been proposed to predict sales at a fast food outlet. ​ $\hat { y } = 18 - 2 x _ { 1 } + 7 x _ { 2 } + 15 x _ { 3 }$ where $x _ { 1 } =$ the number of competitors within 1 mile $x _ { 2 } =$ the population within 1 mile (in $\left. 1000 \mathrm {~s} \right)$ $x _ { 3 } = 1$ if drive-up windows are present; 0 otherwise $\widehat { y } =$ estimated sales $($ in $\$ 1000 \mathrm {~s} )$ a. What is the interpretation of 15 (the coefficient of x₃) in the regression equation? b. Predict sales for an outlet with 2 competitors, a population of 10,000 within one mile, and one drive-up window (give the answer in dollars). c. Predict sales for an outlet with 2 competitors, a population of 10,000 within one mile, and no drive-up window (give the answer in dollars).

As the value of the multiple coefficient of determination increases, ​

In regression analysis, an outlier is an observation whose

The following estimated regression equation was developed relating yearly income (y in $1000s) of 30 individuals with their age (x₁) and their gender (x₂) (0 if male and 1 if female). $\hat { y }$ = 30 + .7x₁ + 3x₂ Also provided are SST = 1200 and SSE = 384.From the above linear function for multiple regression, it can be said that the expected yearly income of

A regression model between sales (y in $1000), 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.To test for the significance of the model, the test statistic F is

A regression model between sales (y in $1000), 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.The coefficient of x₂ indicates that if television advertisement is increased by $1 (holding the unit price constant), sales are expected to

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 $1000s), and type of container used (CONTAIN; 1 = Cans and 0 = Bottles).Part of the regression results is shown below.Here, n = 25. Coefficients Stcanda 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 cost is $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 = .05. c. Test to see if there is a significant relationship between sales and advertising. Let α = .05. d. Is the type of container a significant variable? Let α = .05. e. Test to see if there is a significant relationship between sales and competitor's price. Let α = .05.

The prices (in $) of Rawlston, Inc.stock (y) over a period of 12 days, the number of shares (in 100s) of company's stocks sold (x₁), and the volume of exchange (in millions) on the New York Stock Exchange (x₂) are shown below. Day (y) (x1) (x2) 1 87.50 950 11.00 2 86.00 945 11.25 3 84.00 940 11.75 4 83.00 930 11.75 5 84.50 935 12.00 6 84.00 935 13.00 7 82.00 932 13.25 8 80.00 938 14.50 9 78.50 925 15.00 10 79.00 900 16.50 11 77.00 875 17.00 12 77.50 870 17.50 Excel was used to determine the least squares regression equation.Part of the computer output is shown below. ANOVA df SS MS F Significance F Regression 2 118.8474 59.4237 40.9216 0.0000 Residual 9 13.0692 1.4521 Total 11 131.9167 Coefficients Standard Errort Stat P-value Intercept 118.5059 33.5753 3.5296 0.0064 -0.0163 0.0315 -0.5171 0.6176 -1.5726 0.3590 -4.3807 0.0018 a. $\quad$ Use the output shown above and write an equati on that can be used to predict the price of stock. b. $\quad$ Interpret the coeff cients of the estimated regressi on equation that you found in Part a. c. $\quad$ At a 05 level of significance, determine which variables are significant and which are no If on a given day, the number of shares of the company that were sold was 94,500 and the d. $\quad$ of exchange on the New York Stock Exchange was 16 million, what would you expect th the stock to be?

A regression model between sales (y in $1000), 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.To test for the significance of the model, the p-value is

A multiple regression model has the estimated form $\hat { y }$ = 5 + 6x + 7w As x increases by 1 unit (holding w constant), y is expected to

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 Coefficients 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 to determine 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.

A regression model involving 3 independent variables for a sample of 20 observations resulted in the following sum of squares. Sum of Squares Regression 90 Residual (Error) 100 ​ a. Compute the multiple coefficient of determination and fully explain its meaning. b. At α = .05, test to determine whether or not there is a significant relationship between the independent variables and the dependent variable.

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

A regression analysis was applied to determine the relationship between a dependent variable and 8 independent variables and the following information was obtained. R Square = .80 SSR = 4280 Total number of observations, n = 56 ​ a. Fill in all the blanks in the following ANOVA table. b. Is the model significant? Let α = .05. ​ ANOVA df SS MS F Regression \_\_\_? \_\_\_? \_\_\_? \_\_\_? Error \_\_\_? \_\_\_? \_\_\_? Total \_\_\_? \_\_\_?

Shown below is a partial computer output from a regression analysis. Coefficients Standard Etror Constant 10.00 2.00 -2.00 1.50 6.00 2.00 -4.00 1.00 ANOVA df SS MS F Regression 60 Error Total 19 140 ​ a. Use the above results and write the regression equation. b. Compute the multiple coefficient of determination and fully interpret its meaning. c. At α = .05, test to see if there is a relation between x₁ and y. d. At α = .05, test to see if there is a relation between x₃ and y. e. Is the regression model significant? Perform an F test and let α = .05.

If a categorical variable has k levels, the number of dummy variables required is​

Part of an Excel output relating y (dependent variable) and 4 independent variables, x₁ through x₄, is shown below. ​ Regression Statistics Multiple R ? R Square ? Adjusted R Square 72.6093 Standard Error 20 ANOVA df SS M F S Regression ? 422975.2376 ? ? 0 Residual ? ? ? Total ? ? Coefficients Standard Error t Stat P-value Intercept -203.6125 100.2940 ? 0.0605 0.6483 0.1110 ? 0.0000 0.0190 0.0065 ? 0.0101 40.4577 7.5940 ? 0.0001 -0.1.32 20.7823 ? 0.9961 a. Fill in all the blanks marked with "?". b. At the 5% level of significance, which independent variables are significant and which ones are not? Explain how you arrived at your answers.

The mathematical equation that explains how the dependent variable y is related to several independent variables x₁, x₂, …, xp and the error term ε is