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
To find the equation of the line of best fit, we need to calculate the slope and y-intercept using the given data.
Step 1: Calculate the average of the leg press repetitions and the average 40-yard dash time.
Average leg press repetitions:
(12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 18
Average 40-yard dash time:
(8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 ≈ 10.7
Step 2: Calculate the deviations from the mean for both sets of data.
Leg press deviations:
12 - 18 = -6
32 - 18 = 14
7 - 18 = -11
11 - 18 = -7
23 - 18 = 5
28 - 18 = 10
15 - 18 = -3
40-yard dash time deviations:
8.6 - 10.7 ≈ -2.1
14.6 - 10.7 ≈ 3.9
7.1 - 10.7 ≈ -3.6
8.3 - 10.7 ≈ -2.4
11.9 - 10.7 ≈ 1.2
13.4 - 10.7 ≈ 2.7
9.5 - 10.7 ≈ -1.2
Step 3: Calculate the sum of the products of the deviations for both sets of data.
Sum of (leg press deviation * 40-yard dash deviation):
(-6) * (-2.1) + 14 * 3.9 + (-11) * (-3.6) + (-7) * (-2.4) + 5 * 1.2 + 10 * 2.7 + (-3) * (-1.2)
= 12.6 + 54.6 + 39.6 + 16.8 + 6 + 27 + 3.6
= 160.2
Step 4: Calculate the sum of the squares of the deviations for both sets of data.
Sum of leg press deviations squared:
(-6)^2 + 14^2 + (-11)^2 + (-7)^2 + 5^2 + 10^2 + (-3)^2
= 36 + 196 + 121 + 49 + 25 + 100 + 9
= 536
Sum of 40-yard dash time deviations squared:
(-2.1)^2 + 3.9^2 + (-3.6)^2 + (-2.4)^2 + 1.2^2 + 2.7^2 + (-1.2)^2
= 4.41 + 15.21 + 12.96 + 5.76 + 1.44 + 7.29 + 1.44
= 48.51
Step 5: Calculate the slope of the line of best fit.
slope = sum of (leg press deviation * 40-yard dash deviation) / sum of leg press deviations squared
= 160.2 / 536
= 0.2985
Step 6: Calculate the y-intercept of the line of best fit.
y-intercept = average 40-yard dash time - slope * average leg press repetitions
= 10.7 - 0.2985 * 18
= 10.7 - 5.373
= 5.327
Step 7: Write the equation of the line of best fit.
y = slope * x + y-intercept
y = 0.2985 * x + 5.327
So, the equation of the line of best fit is y = 0.2985x + 5.327.
To find how many seconds a player should expect to take to run 40 yards if they can do 22 leg press repetitions:
y = 0.2985 * 22 + 5.327
y ≈ 11.5717
Therefore, the player should expect to take approximately 11.6 seconds to run 40 yards if they can do 22 leg press repetitions.