A sports medicine major wanted to conduct an experiment to determine if there is a correlation (2 between the members of the soccer team's leg strength and the time it takes for them to sprint 40 yards. He sets up the following test and records the data: Every day for a week he counts how many times each player can leg press 350 pounds. The following week, he has each player sprint 40 yards every day. The table shows the average number of leg-press repetitions and the average 40-yard dash time (in seconds) for seven randomly selected players. What is the equation of the line of best fit? How many seconds should he expect a player to take to run 40 yards if that player can do 22 leg-press repetitions? Round any values to the nearest tenth, if necessary.
leg press (reps) 12, 32, 7, 11, 23, 28, 15
40-yard dash(s) 8.6, 14.6, 7.1, 8.3, 11.9, 13.4, 9.5
To find the equation of the line of best fit, we can use the least squares regression method. We first calculate the mean of the leg press repetitions and the 40-yard dash times:
Mean of leg press reps (X̄) = (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 18.4
Mean of 40-yard dash times (Ȳ) = (8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.3
Next, we find the sums of the products of the differences between each leg press repetition and the mean of leg press reps (Xi - X̄) and each 40-yard dash time and the mean of 40-yard dash times (Yi - Ȳ):
Σ((Xi - X̄)(Yi - Ȳ)) = (12 - 18.4)(8.6 - 10.3) + (32 - 18.4)(14.6 - 10.3) + (7 - 18.4)(7.1 - 10.3) + (11 - 18.4)(8.3 - 10.3) + (23 - 18.4)(11.9 - 10.3) + (28 - 18.4)(13.4 - 10.3) + (15 - 18.4)(9.5 - 10.3) = 324.24
We also need to calculate the sum of the squared differences between each leg press repetition and the mean of leg press reps ((Xi - X̄)^2) and the sum of the squared differences between each 40-yard dash time and the mean of 40-yard dash times ((Yi - Ȳ)^2):
Σ((Xi - X̄)^2) = (12 - 18.4)^2 + (32 - 18.4)^2 + (7 - 18.4)^2 + (11 - 18.4)^2 + (23 - 18.4)^2 + (28 - 18.4)^2 + (15 - 18.4)^2 = 564.8
Σ((Yi - Ȳ)^2) = (8.6 - 10.3)^2 + (14.6 - 10.3)^2 + (7.1 - 10.3)^2 + (8.3 - 10.3)^2 + (11.9 - 10.3)^2 + (13.4 - 10.3)^2 + (9.5 - 10.3)^2 = 17.14
Using these values, we can find the slope of the line of best fit (b):
b = Σ((Xi - X̄)(Yi - Ȳ)) / Σ((Xi - X̄)^2)
b = 324.24 / 564.8 ≈ 0.573
Next, we find the y-intercept of the line of best fit (a):
a = Ȳ - bX̄
a = 10.3 - (0.573 * 18.4) ≈ 0.384
Therefore, the equation of the line of best fit is:
Y ≈ 0.573X + 0.384
To find how many seconds a player should expect to take to run 40 yards if they can do 22 leg press repetitions, we can substitute X = 22 into the equation:
Y ≈ 0.573 * 22 + 0.384 ≈ 12.936
Therefore, the player should expect to take approximately 13 seconds to run 40 yards.