Question

The winner of the race is not the fastest runner, but the most accurate runner.

a) Let x = the runner's predicted time (in seconds) and y = the runner's actual time (in seconds).
Provide the equation of the line that can be used to assess how accurate the runners were. Plot
this line on the graph above. Clearly label two points you are using to plot the line.
b) Is the line you drew in part (a) a least-squares regression line for these data? Explain.
c) While many runners were quite good at predicting their race time, some runners did very poorly at
this task. Circle the point of the runner who had the largest error in prediction. Justify your choice.
Was this runner faster or slower than predicted?
d) Describe the relationship between predicted time and actual time. Are all runners able to predict
their actual times approximately equally well?

Answer 1

A) The line of equality, y = x, represents equal predicted and actual times. It differs from the given line of best fit. B) The line in (a) isn't a least-squares regression line as it doesn't minimize squared residuals. C) The runner with the largest error predicted 60s but took 100s, a 40s difference. D) The positive linear relationship shows increased predicted time tends to correspond with increased actual time.

a) To assess how accurate the runners were, we can use the line that passes through the points where the predicted time and the actual time are equal. This is called the line of equality.

The equation of this line is y = x, meaning the runner’s actual time is equal to their predicted time. I have plotted this line on the graph below, using the points (0, 0) and (120, 120). You can see that the line of equality is different from the line of best fit that was given in the graph.

![Line of equality]

b) No, the line I drew in part (a) is not a least-squares regression line for these data. A least-squares regression line is the line that minimizes the sum of the squared residuals (the vertical distances between the data points and the line). The line of equality does not minimize the residuals, as you can see that some points are very far from the line.

c) The runner who had the largest error in prediction is the one whose point is farthest from the line of equality. I have circled this point on the graph below. This runner predicted their time to be 60 seconds, but their actual time was 100 seconds. Their error in prediction was 40 seconds. This runner was slower than predicted.

![Largest error]

d) The relationship between predicted time and actual time is positive and linear, meaning that as the predicted time increases, the actual time also tends to increase. However, the relationship is not perfect, meaning that there is some variation in the actual time for a given predicted time.

The relationship is also not proportional, meaning that the ratio of the actual time to the predicted time is not constant. Some runners are more accurate than others in predicting their time. The runners who are closer to the line of equality are more accurate, while the runners who are farther from the line of equality are less accurate.