91Ó°ÊÓ

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

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).

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

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

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.

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.