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
To find the line of best fit, we can use the least squares method to calculate the equation of the line that best fits the data points.
First, we need to calculate the means of both sets of data:
Mean of leg press repetitions = (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 18.1
Mean of 40-yard dash time = (8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.4
Next, we calculate the slope of the line of best fit using the formula:
m = Σ(xy) - n*mean(x)*mean(y) / Σ(x^2) - n*(mean(x))^2
= (12*8.6 + 32*14.6 + 7*7.1 + 11*8.3 + 23*11.9 + 28*13.4 + 15*9.5) - 7*18.1*10.4 / (12^2 + 32^2 + 7^2 + 11^2 + 23^2 + 28^2 + 15^2) - 7*(18.1)^2
= 2269.7 - 1300.85 / 2196.75 - 2397.61
= 968.85 / -200.86
= -4.82
Now, we can calculate the y-intercept (b):
b = mean(y) - m*mean(x)
= 10.4 - (-4.82)*18.1
= 10.4 + 87.04
= 97.44
Therefore, the equation of the line of best fit is:
40-yard dash time = -4.82 * leg press repetitions + 97.44
To find out how many seconds a player should take to run 40 yards if they can do 22 leg-press repetitions, we can substitute 22 into the equation:
40-yard dash time = -4.82 * 22 + 97.44
40-yard dash time = -105.64 + 97.44
40-yard dash time = 8.2 seconds
So, a player who can do 22 leg-press repetitions should take approximately 8.2 seconds to run 40 yards.