Question

Exercises 13–28 use the same data sets as Exercises 13–28 in Section 10-1. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 10-5 on page 493.

Use the CPI/subway fare data from the preceding exercise and find

the best predicted subway fare for a time when the CPI reaches 500. What is wrong with this prediction?

Short answer

The regression equation is\(\hat y = - 0.290 + 0.0115x\).

The best-predicted subway fare for a time when the CPI reaches 500 subway fares will be $5.48. The prediction of subway fare at 500 CPI leads to exploitation.

Step-by-step solution

Given information

The given data provides the information of the CPI and subway fare as follows.

State the equation of the regression line

The formula for the estimated regression line is

\(y = {b_0} + {b_1}x\).

Here,

\({b_0}\)is the Y-intercept,

\({b_1}\)is the slope,

\(x\)is the explanatory variable, and

\(\hat y\)is the response variable.

Let X denote the CPI and Y denote the subway fare (in dollars).

Compute the slope and intercept

The calculations required to compute the slope and intercept are as follows.

The sample size is \(\left( n \right) = 9\).

The slope is computed as

\(\begin{array}{c}{b_1} = \frac{{n\left( {\sum {xy} } \right) - \left( {\sum x } \right)\left( {\sum y } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{9 \times 2753.58 - 1427.4 \times 13.85}}{{9 \times 274678.6 - {{1427.4}^2}}}\\ = 0.01153\end{array}\).

The intercept is computed as

\(\begin{array}{c}{b_0} = \frac{{\left( {\sum y } \right)\left( {\sum {{x^2}} } \right) - \left( {\sum x } \right)\left( {\sum {xy} } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{13.85 \times 274678.6 - 1427.4 \times 2753.58}}{{9 \times 274678.6 - {{1427.4}^2}}}\\ = - 0.2903\end{array}\).

Thus, the estimated regression equation is

\(\begin{array}{c}\hat y = {b_0} + {b_1}x\\ = - 0.290 + 0.0115x\end{array}\).

Check the modela

Refer to exercise 16 of section 10-1 for the following result.

1) The scatter plot shows a strong linear relationship between the variables.

2) The P-value is 0.000.

As theP-value is less than the level of significance (0.05), the null hypothesis is rejected.

Therefore, the correlation is statistically significant.

Referring to figure 10-5, the criteria for a good regression model are satisfied.

The regression equation is used to make predictions.

Compute the predicted value

The predicted value of the subway fare when the CPI reaches 500 is

\(\begin{array}{c}\hat y = - 0.290 + 0.0115x\\ = - 0.290 + 0.0115\left( {500} \right)\\ = 5.48\end{array}\).

Thus, the subway fare is predicted as $5.48 when CPI reaches 500.

State the error in the predicted valuea

As the CPI value of500 is not close to the range of values in the sample responses for the variable CPI, the prediction of the subway fare when CPI reaches 500 leads to extrapolation.

Thus, the prediction value is not reliable.