Exam 20: Model Building

Suppose that the sample regression equation of a second-order model is: $= 2.50 + 0.15 x + 0.45 x ^ { 2 }$ . The value 2.50 is the:

The following model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$ + $\varepsilon$ is used whenever the statistician believes that, on average, y is linearly related to:

A professor of accounting wanted to develop a multiple regression model to predict the students' grades in her fourth-year accounting course. She decides that the two most important factors are the student's grade point average (GPA) in the first three years and the student's major. She proposes the model: $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ . where y = fourth-year accounting course mark (out of 100). $x _ { 1 }$ = GPA in first three years (range 0 to 12). $x _ { 2 }$ = 1 if student's major is accounting. = 0 if not. $x _ { 3 }$ = 1 if student's major is finance. = 0 if not. The computer output is shown below. THE REGRESSION EQUATION IS $y =$ $9.14 + 6.73 x _ { 1 } + 10.42 x _ { 2 } + 5.16 x _ { 3 }$ . Predictor Coef StDev T Constant 9.14 7.10 1.287 6.73 1.91 3.524 10.42 4.16 2.505 5.16 3.93 1.313 S = 15.0 R-Sq = 44.2%. ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 17098 5699.333 25.386 Error 96 21553 224.510 Total 99 38651 Do these results allow us to conclude at the 1% significance level that the model is useful in predicting the fourth-year accounting course mark?

In explaining the amount of money spent on children's clothes each month, which of the following independent variables is best represented with an indicator variable?

In a stepwise regression procedure, if two independent variables are highly correlated, then:

A first-order model was used in a regression analysis involving 25 observations to study the relationship between a dependent variable y and three independent variables, $x _ { 1 }$ , $x _ { 2 }$ and $x _ { 3 }$ . The analysis showed that the mean squares for regression is 160 and the sum of squares for error is 1050. In addition, the following is a partial computer printout. Predictor Coef StDev Constant 25 4 18 6 -12 4.8 6 5 Is there sufficient evidence at the 5% significance level to indicate that $x _ { 2 }$ is negatively linearly related to y?

Which of the following best describes when to use an indicator variable in a regression?

A first-order model was used in a regression analysis involving 25 observations to study the relationship between a dependent variable y and three independent variables, $x _ { 1 }$ , $x _ { 2 }$ and $x _ { 3 }$ . The analysis showed that the mean squares for regression is 160 and the sum of squares for error is 1050. In addition, the following is a partial computer printout. Predictor Coef StDev Constant 25 4 18 6 -12 4.8 6 5 Develop the ANOVA table.

In explaining the amount of money spent on children's toys during Christmas each year, the independent variable 'gender' is best represented by a dummy variable.

The model $=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}$ is used whenever the statistician believes that, on average, $y$ is linearly related to $x _ { 1 }$ and $x _ { 2 }$ , and the predictor variables do not interact.

The following model $y = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$ + $\varepsilon$ is referred to as a:

The model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$ + $\varepsilon$ is referred to as a:

In regression analysis, we can use 11 indicator variables to represent 12 months of the year.

A first-order model was used in a regression analysis involving 25 observations to study the relationship between a dependent variable y and three independent variables, $x _ { 1 }$ , $x _ { 2 }$ and $x _ { 3 }$ . The analysis showed that the mean squares for regression is 160 and the sum of squares for error is 1050. In addition, the following is a partial computer printout. Predictor Coef StDev Constant 25 4 18 6 -12 4.8 6 5 Test at the 5% significance level to determine whether $x _ { 1 }$ is linearly related to y.

Suppose that the sample regression line of a first order model is $= 8 + 2 x _ { 1 } + 3 x _ { 2 }$ . If we examine the relationship between y and $x _ { 1 }$ for four different values of $x _ { 2 }$ , we observe that the:

An economist is in the process of developing a model to predict the price of gold. She believes that the two most important variables are the price of a barrel of oil $\left( x _ { 1 } \right)$ and the interest rate $\left( x _ { 2 } \right)$ She proposes the first-order model with interaction: $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } x _ { 3 } + \varepsilon$ . A random sample of 20 daily observations was taken. The computer output is shown below. THE REGRESSION EQUATION IS $y =$ $115.6 + 22.3 x _ { 1 } + 14.7 x _ { 2 } - 1.36 x _ { 1 } x _ { 2 }$ . Predictor Coef SiDev T Constant 115.6 78.1 1.480 22.3 7.1 3.141 14.7 6.3 2.333 -1.36 0.52 -2.615 S = 20.9 R-Sq = 55.4%. ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 8661 2887.0 6.626 Error 16 6971 435.7 Total 19 15632 Is there sufficient evidence at the 1% significance level to conclude that the interest rate and the price of gold are linearly related?

An avid football fan was in the process of examining the factors that determine the success or failure of football teams. He noticed that teams with many rookies and teams with many veterans seem to do quite poorly. To further analyse his beliefs, he took a random sample of 20 teams and proposed a second-order model with one independent variable. The selected model is: $y = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 } + \varepsilon$ . where y = winning team's percentage. x = average years of professional experience. The computer output is shown below: THE REGRESSION EQUATION IS: $y =$ $32.6 + 5.96 x - 0.48 x ^ { 2 }$ Predictor Coef SyDev T Constant 32.6 19.3 1.689 x 5.96 2.41 2.473 -0.48 0.22 -2.182 S = 16.1 R-Sq = 43.9%. ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 2 3452 1726 6.663 Error 17 4404 259.059 Total 19 7856 Test to determine at the 10% significance level if the linear term should be retained.

For the estimated regression equation ŷ = 8 − 5x₁ + 2x₂, which of the following best describes the corresponding change in the value of y, in response to a one unit increase in x₁, while keeping x₂ constant?

The owner of an air conditioner business wants to investigate the relationship between the weekly number of air conditioners sold, temperature and the seasons of the year. A random sample of 14 weeks is taken, with the average temperature of that week (in degrees Celsius) and the quarter from which that week belonged, noted. There are three indicator variables, March, September and December. Excel is used to generate the following multiple linear regression output. SUMMARY OUTPUT Regression Statistics Multiple R 0.99 R Square 0.97 Adjusted RSquare 0.96 Standard Error 4.54 Observations 14.00 ANOVA df SS MS F Significance Regression 4.00 6999.27 1749.82 84.86 0.00 Residual 9.00 185.58 20.62 Total 13.00 7184.86 Coefficients Standard Error tStat P-value Lower 95\% Upper 95\% Intercept -17.94 8.54 -2.10 0.07 -37.27 1.38 Temperature 1.00 0.35 2.84 0.02 0.20 1.79 March 1.01 4.60 0.22 0.83 -9.39 11.40 September 7.22 5.58 1.29 0.23 -5.40 19.84 Deomber 27.87 6.55 4.26 0.00 13.06 42.68 (a) Estimate the number of air conditioners sold in the first week of December, on a 40 degree Celsius day. Is this a good estimate? (b) If the actual number of air conditioners sold in the first week of December was 45 air conditioners, find the residual? Has the model over estimated or underestimated the weekly number of air conditioners sold by this business?

A traffic consultant has analysed the factors that affect the number of traffic fatalities. She has come to the conclusion that two important variables are the number of cars and the number of tractor-trailer trucks. She proposed the second-order model with interaction: $y =$ $\beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } ^ { 2 } + \beta _ { 4 } x _ { 2 } ^ { 2 } + \beta _ { 5 } x _ { 1 } x _ { 2 } + \varepsilon$ . Where: y = number of annual fatalities per shire. $x _ { 1 }$ = number of cars registered in the shire (in units of 10 000). $x _ { 2 }$ = number of trucks registered in the shire (in units of 1000). The computer output (based on a random sample of 35 shires) is shown below. THE REGRESSION EQUATION IS $y =$ $69.7 + 11.3 x _ { 1 } + 7.61 x _ { 2 } - 1.15 x _ { 1 } ^ { 2 } - 0.51 x _ { 2 } ^ { 2 } - 0.13 x _ { 1 } x _ { 2 }$ . Predictor Coef SiDev T Constant 69.7 41.3 1.688 11.3 5.1 2.216 7.61 2.55 2.984 -1.15 0.64 -1.797 -0.51 0.20 -2.55 -0.13 0.10 -1.30 S = 15.2 R-Sq = 47.2%. ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 5 5959 1191.800 5.181 Error 29 6671 230.034 Total 34 12630 Test at the 1% significance level to determine whether the $x _ { 2 }$ term should be retained in the model.