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 can use linear regression. We will use the leg press repetitions as the independent variable (x) and the 40-yard dash time as the dependent variable (y).
First, calculate the mean of the leg press repetitions (x) and the mean of the 40-yard dash times (y).
x̄ = (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 18.1
ȳ = (8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 ≈ 10.5
Next, calculate the sum of the products of each leg press repetition (xi) and 40-yard dash time (yi).
∑(xi * yi) = (12 * 8.6) + (32 * 14.6) + (7 * 7.1) + (11 * 8.3) + (23 * 11.9) + (28 * 13.4) + (15 * 9.5) = 1259.1
Then, calculate the sum of the squares of each leg press repetition (xi) and 40-yard dash time (yi).
∑(xi^2) = (12^2) + (32^2) + (7^2) + (11^2) + (23^2) + (28^2) + (15^2) = 2624
∑(yi^2) = (8.6^2) + (14.6^2) + (7.1^2) + (8.3^2) + (11.9^2) + (13.4^2) + (9.5^2) = 812.61
Now, calculate the slope (m) of the line of best fit:
m = (∑(xi * yi) - (n * x̄ * ȳ)) / (∑(xi^2) - (n * x̄^2))
= (1259.1 - (7 * 18.1 * 10.5)) / (2624 - (7 * 18.1^2))
≈ 0.2642
Finally, calculate the y-intercept (b) of the line of best fit:
b = ȳ - (m * x̄)
= 10.5 - (0.2642 * 18.1)
≈ 5.54
Therefore, the equation of the line of best fit is y = 0.2642x + 5.54.
To find the expected time for a player to run 40 yards if they can do 22 leg press repetitions, we substitute x = 22 into the equation:
y = 0.2642(22) + 5.54
≈ 11.8
Therefore, the player should be expected to take approximately 11.8 seconds to run 40 yards.