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 %C2%A040-yard Dash (s) %C2%A08.6 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.
Using the given data, we can calculate the equation of the line of best fit using the formula:
y = mx + b
Where:
y = 40-yard dash time (in seconds)
x = leg press repetitions
m = slope of the line
b = y-intercept
Using the given data, we calculate the slope (m) and y-intercept (b) using the least squares method:
Step 1: Calculate the means of x-bar (mean of leg press repetitions) and y-bar (mean of 40-yard dash times).
x-bar = (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 18
y-bar = (8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.7
Step 2: Calculate the deviations for x (leg press repetitions) and y (40-yard dash times).
Deviation of x: xi - x-bar
Deviation of y: yi - y-bar
Leg Press Repetitions (x): 12 32 7 11 23 28 15
40-Yard Dash Times (y): 8.6 14.6 7.1 8.3 11.9 13.4 9.5
Deviation of x: -6 14 -11 -7 5 10 -3
Deviation of y: -2.1 3.9 -3.6 -2.4 1.2 2.7 -1.2
Step 3: Calculate the sum of the products between the x and y deviations (xy).
xy = (-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 square of x deviations (x^2).
x^2 = (-6)^2 + (14)^2 + (-11)^2 + (-7)^2 + (5)^2 + (10)^2 + (-3)^2
= 36 + 196 + 121 + 49 + 25 + 100 + 9
= 536
Step 5: Slope (m) can be calculated using the formula:
m = xy / x^2
m = 160.2 / 536
m ≈ 0.2996
Step 6: Y-intercept (b) can be calculated using the formula:
b = y-bar - m * x-bar
b = 10.7 - (0.2996 * 18)
b ≈ 5.3
Therefore, the equation of the line of best fit is:
y = 0.2996x + 5.3
To estimate the 40-yard dash time (y) for a player who can do 22 leg-press repetitions (x), we substitute x = 22 into the equation and solve for y:
y = 0.2996 * 22 + 5.3
y ≈ 11.6
Therefore, the player should expect to take approximately 11.6 seconds to run 40 yards.