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 l
To find the equation of the line of best fit, we can use the linear regression equation:
y = mx + b
Where y is the dependent variable (40-yard dash time), x is the independent variable (leg press repetitions), m is the slope of the line, and b is the y-intercept.
Using the given data points, we can calculate the slope (m) and y-intercept (b) to find the equation of the line.
First, we need to find the means of both the leg press repetitions and 40-yard dash times.
Mean of leg press repetitions (x̄):
(12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 18.1
Mean of 40-yard dash times (ȳ):
(8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.9
Next, we can calculate the slope (m) using the formula:
m = ∑((x - x̄)(y - ȳ)) / ∑((x - x̄)^2)
Let's calculate the numerator first:
((12 - 18.1)(8.6 - 10.9)) + ((32 - 18.1)(14.6 - 10.9)) + ((7 - 18.1)(7.1 - 10.9)) + ((11 - 18.1)(8.3 - 10.9)) + ((23 - 18.1)(11.9 - 10.9)) + ((28 - 18.1)(13.4 - 10.9)) + ((15 - 18.1)(9.5 - 10.9)) = -115.94
Next, let's calculate the denominator:
((12 - 18.1)^2) + ((32 - 18.1)^2) + ((7 - 18.1)^2) + ((11 - 18.1)^2) + ((23 - 18.1)^2) + ((28 - 18.1)^2) + ((15 - 18.1)^2) = 1200.14
Now, we can calculate the slope (m):
m = -115.94 / 1200.14 ≈ -0.0966
Next, we can calculate the y-intercept (b) using the formula:
b = ȳ - mx̄
b = 10.9 - (-0.0966)(18.1) ≈ 12.7
Therefore, the equation of the line of best fit is:
y ≈ -0.097x + 12.7
To find how many seconds a player should expect to take to run 40 yards if they can do 22 leg press repetitions, we substitute x = 22 into the equation:
y = -0.097(22) + 12.7 ≈ 10.1
Therefore, a player who can do 22 leg press repetitions should expect to take approximately 10.1 seconds to run 40 yards.