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 (s)
8.6
14.6
7.1
8.3
11.9
13.4
9.5
(2 points)
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To find the equation of the line of best fit, we can use linear regression analysis. First, we need to calculate the means of the leg press repetitions and 40-yard dash times:
Mean leg press repetitions = (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 17.1
Mean 40-yard dash time = (8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.7
Next, we need to calculate the covariance and variance of the data:
Covariance = Σ((Leg Press - Mean Leg Press) * (40-Yard Dash - Mean Dash)) / n
= [(12-17.1)*(8.6-10.7) + (32-17.1)*(14.6-10.7) + (7-17.1)*(7.1-10.7) + (11-17.1)*(8.3-10.7) + (23-17.1)*(11.9-10.7) + (28-17.1)*(13.4-10.7) + (15-17.1)*(9.5-10.7)] / 7
= [-27.9 * -2.1 + 14.9 * 3.9 + -10.1 * -3.6 + -6.1 * -2.4 + 5.9 * 1.2 + 10.9 * 2.7 + -2.1 * -1.2] / 7
= [58.59 + 58.11 + 36.36 + 14.64 + 7.08 + 29.43 + 2.52] / 7
= 206.73 / 7
= 29.53
Variance of Leg Press = Σ((Leg Press - Mean Leg Press)^2) / n
= [(12-17.1)^2 + (32-17.1)^2 + (7-17.1)^2 + (11-17.1)^2 + (23-17.1)^2 + (28-17.1)^2 + (15-17.1)^2] / 7
= [28.41 + 217.16 + 103.62 + 36.36 + 34.56 + 126.81 + 4.84] / 7
= 551.76 / 7
= 78.82
Now we can calculate the slope of the line of best fit:
Slope = Covariance / Variance of Leg Press
= 29.53 / 78.82
= 0.374
Next, we can calculate the y-intercept of the line of best fit:
y-intercept = Mean Dash - Slope * Mean Leg Press
= 10.7 - 0.374 * 17.1
= 10.7 - 6.4
= 4.3
Therefore, the equation of the line of best fit is:
40-Yard Dash = 0.374 * Leg Press + 4.3
To determine how many seconds a player would take to run 40 yards if they can do 22 leg press repetitions:
40-Yard Dash = 0.374 * 22 + 4.3
= 8.228 + 4.3
= 12.528
Rounded to the nearest tenth, the player would be expected to take approximately 12.5 seconds to run 40 yards.