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 pizza costs and subway fares to find the best predicted

subway fare, given that the cost of a slice of pizza is $3.00. Is the best predicted subway fare likely to be implemented?

Short answer

The regression equation is\(\hat y = - 0.0111 + 1.01x\).

The best-predicted ‘subway fare’ for the cost of a slice of pizza is $3.00 will be approximately $3.02. The best-predicted subway fare of $3.02 is not likely to be implemented due to its convenience to use in a real situation.

Step-by-step solution

Given information

The given data provides the information of the pizza cost (in dollars) and subway fare as follows.

State the estimated 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 (predicted value).

Let X denotes the cost of a pizza slice (in dollars) 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 follows.

\(\begin{array}{c}{b_1} = \frac{{n\left( {\sum {xy} } \right) - \left( {\sum x } \right)\left( {\sum y } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{9 \times 27.8325 - 13.8 \times 13.85}}{{9 \times 27.685 - {{13.8}^2}}}\\ = 1.010856\end{array}\).

The intercept is computed as follows.

\(\begin{array}{c}{b_0} = \frac{{\left( {\sum y } \right)\left( {\sum {{x^2}} } \right) - \left( {\sum x } \right)\left( {\sum {xy} } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{13.8 \times 27.685 - 13.8 \times 27.8325}}{{9 \times 27.685 - {{13.8}^2}}}\\ = - 0.01109\end{array}\).

Thus, the estimated regression equation is

\(\begin{array}{c}\hat y = {b_0} + {b_1}x\\ = - 0.011 + 1.012x\end{array}\).

Check the model

Refer to exercise 15 of section 10-1 for the following result.

1) The scatter plot shows an approximate linear relationship between the variables.

2)The P-value is 0.000.

As the P-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.

Thus, the prediction is made using a regression equation.

Compute the prediction

The best-predicted subway fare for the cost of a slice of pizza of $3.00 is required.

Therefore, the estimated value for $3.00 is

\(\begin{array}{c}\hat y = {b_0} + {b_1}x\\ = - 0.0111 + 1.01x\\ = - 0.0111 + 1.01 \times 3\\ \approx 3.02\end{array}\).

Therefore, the best-predicted subway fare for the cost of a slice of pizza, which is $3.00, will be approximately $3.02.

The prediction of $3.02 is not likely to be possible as the denomination is not a convenient value, as compared to values like $3.25 or $3.00.