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): ${\hat{\mathbf{\beta }}}_{\mathbf{2}}$

a. Explain why the value${\hat{\mathbf{\beta }}}_{\mathbf{0}}\mathbf{=}\mathbf{325790}$has no practical interpretation.

b. Explain why the value${\hat{\mathbf{\beta }}}_{\mathbf{1}}\mathbf{=}\mathbf{-}\mathbf{321}\mathbf{.}\mathbf{67}$should not be Interpreted as a slope.

c. Examine the value of${\hat{\mathbf{\beta }}}_{\mathbf{2}}$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.)

Short answer

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${\hat{\mathbf{\beta }}}_{\mathbf{2}}$ 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

Interpretation of β0

Here, the value of ${\hat{\mathbf{\beta }}}_{\mathbf{1}}$is 325,790. ${\mathbf{\beta }}_{\mathbf{0}}$represent 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.

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.

Clarification of β2

The value of${\hat{\mathbf{\beta }}}_{\mathbf{2}}$ is coming out to be 0.0794. A positive value here denotes that the curve is upward sloping.

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.