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Chapter 12: Q56E. (page 747)

Catalytic converters in cars. A quadratic model was applied to motor vehicle toxic emissions data collected in Mexico City (Environmental Science & Engineering, Sept. 1, 2000). The following equation was used to predict the percentage (y) of motor vehicles without catalytic converters in the Mexico City fleet for a given year (x): 尾^2

a. Explain why the value尾^0=325790has no practical interpretation.

b. Explain why the value尾^1=-321.67should not be Interpreted as a slope.

c. Examine the value of尾^2to determine the nature of the curvature (upward or downward) in the sample data.

d. The researchers used the model to estimate 鈥渢hat just after the year 2021 the fleet of cars with catalytic converters will completely disappear.鈥� Comment on the danger of using the model to predict y in the year 2021. (Note: The model was fit to data collected between 1984 and 1999.)

Short Answer

Expert verified

a. Since, our x variable is a time concept it cannot be zero, hence, we cannot practically interpret the value.

b. The percentage of motor vehicles (y) without catalytic converters for a given year (x) is predicted here. When x goes up by 1-unit, y according to the question decreases by 321.67 units. But since we are measuring y in percentage form this number is not reliable and thus should not be interpreted as a slope.

c. The value of尾^2 is coming out to be 0.0794. A positive value here denotes that the curve is upward sloping.

d. The model was fit to data collected between 1984 and 1999. The researchers want to use the regression equation to predict y in the year 2021. Using the model to predict y in the year 2021 will not give very accurate results as almost 20 years have passed and the variables and their relationship with y changes over due period of time.

Step by step solution

01

Interpretation of β0

Here, the value of 尾^1is 325,790. 尾0represent the y-intercept and the value here 325,790 denotes no of motor vehicles without catalytic converters in the Mexico City fleet for a given year. Since, our x variable is a time concept it cannot be zero, hence, we cannot practically interpret the value.

02

Simplification of β1

The percentage of motor vehicles (y) without catalytic converters for a given year (x) is predicted here. When x goes up by 1-unit, y according to the question decreases by 321.67 units. But since we are measuring y in percentage form this number is not reliable and thus should not be interpreted as a slope.

03

Clarification of β2

The value of尾^2 is coming out to be 0.0794. A positive value here denotes that the curve is upward sloping.

04

Prediction

The model was fit to data collected between 1984 and 1999. The researchers want to use the regression equation to predict y in the year 2021. Using the model to predict y in the year 2021 will not give very accurate results as almost 20 years have passed and the variables and their relationship with y changes overdue period of time.

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Most popular questions from this chapter

Question: Job performance under time pressure. Refer to the Academy of Management Journal (October 2015) study of how time pressure affects team job performance, Exercise 12.89 (p. 765). Recall that the researchers hypothesized a complete second-order model relating team performance (y) to perceived time pressure (x1), and whether or not the team had an effective leader (x2 = 1 if yes, 0 if no):

E(Y)=尾0+尾1x1+尾2x22+尾3x2+尾4x1x2+尾5x12x2

a) How would you determine whether the rate of increase of team performance with time pressure depends on effectiveness of the team leader?

b) For fixed time pressure, how would you determine whether the mean team performance differs for teams with effective and non-effective team leaders?

Going for it on fourth down in the NFL. Refer to the Chance (Winter 2009) study of fourth-down decisions by coaches in the National Football League (NFL), Exercise 11.69 (p. 679). Recall that statisticians at California State University, Northridge, fit a straight-line model for predicting the number of points scored (y) by a team that has a first-down with a given number of yards (x) from the opposing goal line. A second model fit to data collected on five NFL teams from a recent season was the quadratic regression model, E(y)=尾0+尾1x+尾2x2.The regression yielded the following results: y=6.13+0.141x-0.0009x2,R2=0.226.

a) If possible, give a practical interpretation of each of the b estimates in the model.

b) Give a practical interpretation of the coefficient of determination,R2.

c) In Exercise 11.63, the coefficient of correlation for the straight-line model was reported asR2=0.18. Does this statistic alone indicate that the quadratic model is a better fit than the straight-line model? Explain.

d) What test of hypothesis would you conduct to determine if the quadratic model is a better fit than the straight-line model?

Question: Identify the problem(s) in each of the residual plots shown below.

Question: Write a regression model relating E(y) to a qualitative independent variable that can assume three levels. Interpret all the terms in the model.

Question: There are six independent variables, x1, x2, x3, x4, x5, and x6, that might be useful in predicting a response y. A total of n = 50 observations is available, and it is decided to employ stepwise regression to help in selecting the independent variables that appear to be useful. The software fits all possible one-variable models of the form

E(Y)=尾0+尾1xi

where xi is the ith independent variable, i = 1, 2, 鈥�, 6. The information in the table is provided from the computer printout.

a. Which independent variable is declared the best one variable predictor of y? Explain.

b. Would this variable be included in the model at this stage? Explain.

c. Describe the next phase that a stepwise procedure would execute.
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