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)
To find the equation of the line of best fit, we need to calculate the slope and y-intercept.
First, we need to find the average number of leg-press repetitions and the average 40-yard dash time.
Average leg-press repetitions = (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 17.9
Average 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 find the deviation for each data point.
Leg-press repetitions:
12 - 17.9 = -5.9
32 - 17.9 = 14.1
7 - 17.9 = -10.9
11 - 17.9 = -6.9
23 - 17.9 = 5.1
28 - 17.9 = 10.1
15 - 17.9 = -2.9
40-yard dash time:
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
Next, we need to find the square of the deviations.
Leg-press repetitions:
(-5.9)^2 = 34.81
(14.1)^2 = 198.81
(-10.9)^2 = 118.81
(-6.9)^2 = 47.61
(5.1)^2 = 26.01
(10.1)^2 = 102.01
(-2.9)^2 = 8.41
40-yard dash time:
(-2.1)^2 = 4.41
(3.9)^2 = 15.21
(-3.6)^2 = 12.96
(-2.4)^2 = 5.76
(1.2)^2 = 1.44
(2.7)^2 = 7.29
(-1.2)^2 = 1.44
Next, we need to find the products of the deviations.
(-5.9)(-2.1) = 12.39
(14.1)(3.9) = 54.99
(-10.9)(-3.6) = 39.24
(-6.9)(-2.4) = 16.56
(5.1)(1.2) = 6.12
(10.1)(2.7) = 27.27
(-2.9)(-1.2) = 3.48
Next, we need to find the sum of the squares of the deviations.
Sum of squares of leg-press repetitions = 34.81 + 198.81 + 118.81 + 47.61 + 26.01 + 102.01 + 8.41 = 536.66
Sum of squares of 40-yard dash time = 4.41 + 15.21 + 12.96 + 5.76 + 1.44 + 7.29 + 1.44 = 48.51
Next, we need to find the sum of the products of the deviations.
Sum of products of leg-press repetitions and 40-yard dash time = 12.39 + 54.99 + 39.24 + 16.56 + 6.12 + 27.27 + 3.48 = 160.05
Next, we can calculate the slope.
Slope = (sum of products of deviations) / (sum of squares of leg-press repetitions) = 160.05 / 536.66 ≈ 0.298
Finally, we can calculate the y-intercept.
y-intercept = (average 40-yard dash time) - (slope) * (average leg-press repetitions) = 10.7 - 0.298 * 17.9 ≈ 10.7 - 5.327 ≈ 5.4
Therefore, the equation of the line of best fit is:
y = 0.298x + 5.4
To find how many seconds a player should expect to take to run 40 yards if that player can do 22 leg-press repetitions, we can substitute x = 22 into the equation:
y = 0.298(22) + 5.4 = 6.556 + 5.4 ≈ 12.0
Therefore, the player should expect to take approximately 12.0 seconds to run 40 yards.