Catalytic converters in cars. A quadratic model was

Chapter 12, Problem 56E

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QUESTION:

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):

\(\hat{y}=325,790-321.67 x+.0794 x^{2}\)

a. Explain why the value \(\hat{\beta}_{0}=325,790\) has no practical interpretation.

b. Explain why the value \(\hat{\beta}_{1}=-321.67\) should not be interpreted as a slope.

c. Examine the value of \(\hat{\beta}_{2}\) to determine the nature of the curvature (upward or downward) in the sample data.

d. The researchers used the model to estimate "that just after the year 2021 the fleet of cars with catalytic converters will completely disappear." 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.)

Text Transcription:

hat{y} = 325,790 - 321.67x + .0794 x^2

hat{beta}_0 = 325,790

hat{beta}_1 = -321.67

Questions & Answers

QUESTION:

\(\hat{y}=325,790-321.67 x+.0794 x^{2}\)

a. Explain why the value \(\hat{\beta}_{0}=325,790\) has no practical interpretation.

b. Explain why the value \(\hat{\beta}_{1}=-321.67\) should not be interpreted as a slope.

c. Examine the value of \(\hat{\beta}_{2}\) to determine the nature of the curvature (upward or downward) in the sample data.

Text Transcription:

hat{y} = 325,790 - 321.67x + .0794 x^2

hat{beta}_0 = 325,790

hat{beta}_1 = -321.67

ANSWER:

Step 1 of 4

(a)

Interpretation:

In the given model, , that is the independent variables cannot predict whether the values of variables increase or decrease. Thus, the estimated values remain constant (325,790) in the model.