Question

Ap statistics teachers from across the country meet annually in Kansas City to score Ap statistics exams. One social event that many teachers participate in is the annual prediction fun run. Prior to the race the teachers predict how long it will take them to complete the 1.96 mile long course. During the event, they are timed and their actual finishing times are paired with their predicted times. Here is a scatter plot of the results. The winner of the race is not the fastest runner but the most accurate runner.

i) let x=runners 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 used to plot the line.
ii) Is the line you drew in part a) a least squares regression line for these data? Explain.
iii) 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?

Answer 1

The equation to assess the runners' accuracy is y = x, which is not a least squares regression line. To find the runner with the largest error, we look for the point farthest from this line on the scatter plot. The position of the point relative to the y = x line indicates if the runner was faster or slower than predicted.

Step-by-step explanation:

To assess the accuracy of the runners in predicting their race times, the line we would plot is the line where the predicted time (x) equals the actual time (y). Hence, the equation of the line would be y = x. This line would be drawn too by picking any two points where the predicted time and the actual time are equal. For example, if a runner predicted they would take 500 seconds and actually took 500 seconds, one point would be (500, 500) on the graph.

The line y = x is not a least squares regression line; it's simply a line where prediction equals reality. The least squares regression line, however, would be calculated through a statistical method to minimize the sum of squared residuals between the actual data points and the predicted points on the line.

To find the runner with the largest error in prediction, we would look for the point with the greatest distance from the y = x line. The direction from the line indicates if the runner was faster or slower than predicted. If the point is above the line, the runner was slower than predicted (actual time y was greater than predicted time x). If it's below, the runner was faster.