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 for this data, we can use linear regression. Let's begin by calculating the mean of the leg press repetitions and the 40-yard dash times:
Mean of leg press repetitions:
(12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 18
Mean of 40-yard dash times:
(8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 ≈ 10.8
Next, we need to calculate the deviations of each data point from the mean:
Leg press repetitions 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 - 10.8 ≈ -2.2
14.6 - 10.8 ≈ 3.8
7.1 - 10.8 ≈ -3.7
8.3 - 10.8 ≈ -2.5
11.9 - 10.8 ≈ 1.1
13.4 - 10.8 ≈ 2.6
9.5 - 10.8 ≈ -1.3
Next, we can calculate the product of the deviations for each player:
(-6) * (-2.2) = 13.2
14 * 3.8 = 53.2
(-11) * (-3.7) = 40.7
(-7) * (-2.5) = 17.5
5 * 1.1 = 5.5
10 * 2.6 = 26
(-3) * (-1.3) = 3.9
We also need to calculate the square of each deviation:
(-6)^2 = 36
14^2 = 196
(-11)^2 = 121
(-7)^2 = 49
5^2 = 25
10^2 = 100
(-3)^2 = 9
Now, we can calculate the sum of the products and the sum of the squared deviations:
Sum of the products = 13.2 + 53.2 + 40.7 + 17.5 + 5.5 + 26 + 3.9 = 160.0
Sum of the squared deviations = 36 + 196 + 121 + 49 + 25 + 100 + 9 = 536
Next, we can calculate the slope of the line:
Slope (b) = Sum of the products / Sum of the squared deviations
Slope = 160.0 / 536 ≈ 0.2985
Now, let's find the y-intercept (a) using the formula:
Mean of the 40-yard dash times = a + b * Mean of leg press repetitions
10.8 = a + 0.2985 * 18
We can solve for a:
a + 0.2985 * 18 = 10.8
a + 5.373 ≈ 10.8
a ≈ 10.8 - 5.373
a ≈ 5.427
Therefore, the equation of the line of best fit is:
40-yard dash time = 5.427 + 0.2985 * leg press repetitions
Now, let's use this equation to predict the 40-yard dash time for a player who can do 22 leg press repetitions:
40-yard dash time = 5.427 + 0.2985 * 22
40-yard dash time ≈ 5.427 + 6.566
40-yard dash time ≈ 11.0
Therefore, the player who can do 22 leg press repetitions is expected to take approximately 11 seconds to run 40 yards.