A sports medicine major wanted to conduct an experiment to determine if there is a correlation 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 - 8.6, 14.6, 7.1, 8.3, 11.9, 13.4, 9.5
First, let's calculate the mean of the leg press repetitions and 40-yard dash times:
Mean of leg press reps = (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 18.1
Mean of 40-yard dash times = (8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.9
Next, we will calculate the slope of the line of best fit using the formula:
Slope (m) = Σ [(X - X̅) * (Y - Ȳ)] / Σ (X - X̅)^2
where Σ represents the sum, X is the leg press repetitions, X̅ is the mean of leg press reps, Y is the 40-yard dash time, and Ȳ is the mean of 40-yard dash times.
Using the data given, we calculate the slope as follows:
((12-18.1) * (8.6-10.9) + (32-18.1) * (14.6-10.9) + (7-18.1) * (7.1-10.9) + (11-18.1) * (8.3-10.9) + (23-18.1) * (11.9-10.9) + (28-18.1) * (13.4-10.9) + (15-18.1) * (9.5-10.9)) / ((12-18.1)^2 + (32-18.1)^2 + (7-18.1)^2 + (11-18.1)^2 + (23-18.1)^2 + (28-18.1)^2 + (15-18.1)^2) = 1.041
Next, we can calculate the y-intercept (b) using the formula:
b = Ȳ - m * X̅
b = 10.9 - 1.041 * 18.1 = -7.3
Therefore, the equation of the line of best fit is:
Y = 1.041X - 7.3
If a player can do 22 leg-press repetitions, then using the equation above:
Y = 1.041 * 22 - 7.3 = 14.2
Therefore, we should expect the player to take approximately 14.2 seconds to run 40 yards.