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:%0D%0A%0D%0AEvery 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.
To find the equation of the line of best fit, we need to perform a linear regression analysis on the data. Here is the calculation:
Using the given data:
Average number of leg-press repetitions (x): 15, 18, 20, 22, 25, 27, 30
Average 40-yard dash time (y): 5.2, 4.9, 4.8, 4.6, 4.4, 4.3, 4.1
1. Calculate the mean of x (x̄) and y (ȳ):
x̄ = (15 + 18 + 20 + 22 + 25 + 27 + 30) / 7 = 21.4
ȳ = (5.2 + 4.9 + 4.8 + 4.6 + 4.4 + 4.3 + 4.1) / 7 = 4.6
2. Calculate the deviations from the mean for x (xi - x̄) and y (yi - ȳ):
xi - x̄: -6.4, -3.4, -1.4, 0.6, 3.6, 5.6, 8.6
yi - ȳ: 0.6, 0.3, 0.2, -0.4, -0.2, -0.3, -0.5
3. Calculate the product of the deviations for each data point (xi - x̄)(yi - ȳ):
(-6.4 * 0.6), (-3.4 * 0.3), (-1.4 * 0.2), (0.6 * -0.4), (3.6 * -0.2), (5.6 * -0.3), (8.6 * -0.5) = -3.84, -1.02, -0.28, -0.24, -0.72, -1.68, -4.30
4. Calculate the squared deviations for x [(xi - x̄)^2]:
(-6.4)^2, (-3.4)^2, (-1.4)^2, (0.6)^2, (3.6)^2, (5.6)^2, (8.6)^2 = 40.96, 11.56, 1.96, 0.36, 12.96, 31.36, 73.96
5. Calculate the sum of the squared deviations for x [(xi - x̄)^2]:
40.96 + 11.56 + 1.96 + 0.36 + 12.96 + 31.36 + 73.96 = 172.12
6. Calculate the sum of the product of the deviations for each data point:
-3.84 + -1.02 + -0.28 + -0.24 + -0.72 + -1.68 + -4.30 = -12.08
7. Calculate the slope (b) of the line of best fit using the formula:
b = Σ[(xi - x̄)(yi - ȳ)] / Σ[(xi - x̄)^2]
b = -12.08 / 172.12 ≈ -0.0702
8. Calculate the y-intercept (a) of the line of best fit using the formula:
a = ȳ - b * x̄
a = 4.6 - (-0.0702 * 21.4) ≈ 6.05
9. Write the equation of the line of best fit in the form: y = a + bx
y = 6.05 - 0.0702x
So, the equation of the line of best fit is y = 6.05 - 0.0702x.
To find the expected 40-yard dash time for a player who can do 22 leg-press repetitions, substitute x = 22 into the equation and solve for y:
y = 6.05 - 0.0702(22)
y ≈ 4.47
Therefore, the player should be expected to take approximately 4.5 seconds to run 40 yards.