Question

The total race time for a 100-meter dash can be considered the sum of two variables: the reaction time to the starting signal and the running time for the 100 meters. The scatterplot shows reaction times and running times for 20 runners in a certain race. The winner was the runner with the least total race time.
(a) Circle the point on the graph that represents the runner who won the race and approximate the total race time for that runner.

(b) Based on the graph, is it reasonable to assume that reaction time and running time are independent? Justify your answer.

(c) Based on the least-squares regression model created from the data, explain why the use of extrapolation to predict the running time for a runner whose reaction time is 0.30 second might not be appropriate.

Answer 1

Answer:

(a)  The runner who won the race is the runner with the least running time of 9.6 seconds

(b) The reaction time and running time are proportional on average

(c) Long range assumption of least-squares regression

Poor sensitivity to outliers

Poor extrapolation characterization

Explanation:

We are told that

Total race time = Reaction time + Running Time

Winner = Runner with least total race time =  Runner with least (Reaction time + Running Time)

From the scatter plot, therefore, the winner has the lowest sum of x + y values, therefore, the winner will be located close to the origin.

The three closest to the origin have x + y values,

0.135 + 9.7 = 9.835

0.145 + 9.7 = 9.845

0.153 + 9.6 = 9.753

There, the runner who won the race is the runner with the least running time of 9.6 seconds

(b) No because, to a slight approximation, as the running time increases, the reaction time also increases

(c) As a reaction time of 0.3 second is an outlier, a least-square regression model would be inappropriate due to the long range, poor sensitivity to outliers and poor extrapolation characterization.