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
To find the equation of the line of best fit, we can use linear regression. However, since we do not have access to a calculator or software with regression capabilities, I will show you how to do it manually using the least squares method.
First, let's calculate the means of both the leg press repetitions and the 40-yard dash times:
Mean of leg press repetitions:
(12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 18.1
Mean of 40-yard dash times:
(8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.9
Now, let's calculate the differences between each data point and the mean for both variables:
Leg press repetitions differences:
(12 - 18.1, 32 - 18.1, 7 - 18.1, 11 - 18.1, 23 - 18.1, 28 - 18.1, 15 - 18.1) = (-6.1, 13.9, -11.1, -7.1, 4.9, 9.9, -3.1)
40-yard dash time differences:
(8.6 - 10.9, 14.6 - 10.9, 7.1 - 10.9, 8.3 - 10.9, 11.9 - 10.9, 13.4 - 10.9, 9.5 - 10.9) = (-2.3, 3.7, -3.8, -2.6, 1.0, 2.5, -1.4)
Next, we need to calculate the product of each pair of differences:
Product of differences:
(-6.1 * -2.3, 13.9 * 3.7, -11.1 * -3.8, -7.1 * -2.6, 4.9 * 1.0, 9.9 * 2.5, -3.1 * -1.4) = (14.03, 51.43, -42.18, 18.46, 4.90, 24.75, -4.34)
Now, let's calculate the sum of these products:
14.03 + 51.43 + (-42.18) + 18.46 + 4.9 + 24.75 + (-4.34) = 63.95
Next, let's calculate the sum of the squared leg press repetitions differences:
(-6.1^2 + 13.9^2 + -11.1^2 + -7.1^2 + 4.9^2 + 9.9^2 + -3.1^2) = (37.21 + 193.21 + 123.21 + 50.41 + 24.01 + 98.01 + 9.61) = 535.47
Finally, let's calculate the slope of the line using these values:
Slope = Sum of product of differences / Sum of squared leg press repetitions differences
Slope = 63.95 / 535.47
Slope ≈ 0.1195
Now that we have the slope, we can use the means to find the y-intercept:
y-intercept = Mean 40-yard dash times - Slope * Mean leg press repetitions
y-intercept = 10.9 - 0.1195 * 18.1
y-intercept ≈ 8.75
Therefore, the equation of the line of best fit is:
y = 0.1195x + 8.75
To find the expected 40-yard dash time for a player who can do 22 leg-press repetitions, we substitute x = 22 into the equation:
y = 0.1195 * 22 + 8.75
y ≈ 11.59
Therefore, we should expect a player who can do 22 leg-press repetitions to take approximately 11.59 seconds to run 40 yards.