Use the time/tip data from the table below, which includes data from New York City taxi rides. (The distances are in miles, the times are in minutes, the fares are in dollars, and the tips are in dollars.) Find the regression equation, letting time be the predictor [tex](x)[/tex] variable. Find the best predicted tip for a ride that takes 30 minutes. How does the result compare to the actual tip amount of [tex]\$4.70[/tex]? Use a significance level of 0.05:

\begin{tabular}{l|cccccccc}
Distance & 8.51 & 1.80 & 12.71 & 2.47 & 1.02 & 1.32 & 0.49 & 1.65 \\
\hline Time & 31.00 & 25.00 & 27.00 & 18.00 & 8.00 & 8.00 & 2.00 & 11.00 \\
\hline Fare & 31.75 & 16.30 & 36.80 & 14.30 & 7.80 & 7.80 & 4.80 & 9.80 \\
\hline Tip & 2.98 & 1.50 & 0.00 & 4.29 & 2.34 & 0.00 & 0.00 & 1.96
\end{tabular}

The regression equation is [tex]\hat{y} = 2.63 + 0.2467x[/tex] (Round the [tex]y[/tex]-intercept to two decimal places as needed. Round the slope to four decimal places as needed.)

The best predicted tip for a ride that takes 30 minutes is [tex]\$ 10.03[/tex]. (Round to the nearest cent as needed.)

How does the result compare to the actual tip amount of [tex]\$ 4.70[/tex]?

A. The best predicted value is very different from the actual tip of [tex]\$ 4.70[/tex].
B. The best predicted value is exactly the same as the actual tip of [tex]\$ 4.70[/tex].
C. The best predicted value is close to the actual tip of [tex]\$ 4.70[/tex].
D. The result does not make sense given the context of the data.



Answer :

To address the given problem, we will use the regression equation to predict the tip for a taxi ride that takes 30 minutes and compare this predicted tip to the actual tip of $4.70. Let’s go through the solution step-by-step.

1. Identify the provided regression equation:
The regression equation given is:
[tex]\[ \hat{y} = 24.63 + 0.2467 \times \text{time} \][/tex]

2. Substitute the specified time (30 minutes) into the regression equation:
[tex]\[ \hat{y} = 24.63 + 0.2467 \times 30 \][/tex]

3. Calculate the predicted tip:
[tex]\[ \hat{y} = 24.63 + (0.2467 \times 30) \][/tex]
First, multiply the slope (0.2467) by the time (30 minutes):
[tex]\[ 0.2467 \times 30 = 7.401 \][/tex]
Then, add this result to the y-intercept (24.63):
[tex]\[ \hat{y} = 24.63 + 7.401 = 32.031 \][/tex]
Thus, the best-predicted tip for a ride that takes 30 minutes is approximately $32.03 when rounded to the nearest cent.

4. Comparison to the actual tip of $4.70:
- The predicted tip is $32.03.
- The actual tip is $4.70.

To decide whether the predicted tip is very different, exactly the same, or close to the actual tip:
- If the predicted tip were exactly $4.70, it would be exactly the same. This is not the case.
- If the predicted tip were within [tex]$0.50 of $[/tex]4.70 (i.e., between [tex]$4.20 and $[/tex]5.20), it would be considered close. This is also not the case.
- Since [tex]$32.03 is far from $[/tex]4.70, we conclude that the best-predicted value is very different from the actual tip of $4.70.

Therefore, the correct comparison according to the given criteria is:
[tex]\[ \boxed{\text{A. The best predicted value is very different from the actual tip of \$4.70.}} \][/tex]