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: 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 the method of least squares to find the equation for a straight line that comes closest to fitting the data points.
First, we calculate the mean values for the leg press repetitions and the 40-yard dash times:
Mean of leg press repetitions (x̄) = (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 17.9
Mean of 40-yard dash times (ȳ) = (8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.8
Next, we calculate the deviations from the mean for both variables:
Deviation of leg press repetitions (x) = (12 - 17.9, 32 - 17.9, 7 - 17.9, 11 - 17.9, 23 - 17.9, 28 - 17.9, 15 - 17.9) = (-5.9, 14.1, -10.9, -6.9, 5.1, 10.1, -2.9)
Deviation of 40-yard dash times (y) = (8.6 - 10.8, 14.6 - 10.8, 7.1 - 10.8, 8.3 - 10.8, 11.9 - 10.8, 13.4 - 10.8, 9.5 - 10.8) = (-2.2, 3.8, -3.7, -2.5, 1.1, 2.6, -1.3)
Next, we calculate the product of the deviations:
Product of deviations (xy) = (-5.9 * -2.2, 14.1 * 3.8, -10.9 * -3.7, -6.9 * -2.5, 5.1 * 1.1, 10.1 * 2.6, -2.9 * -1.3) = (12.98, 53.58, 40.33, 17.25, 5.61, 26.26, -3.77)
We also calculate the squared deviations for the leg press repetitions:
Squared deviation for leg press repetitions (x^2) = (-5.9)^2 + (14.1)^2 + (-10.9)^2 + (-6.9)^2 + (5.1)^2 + (10.1)^2 + (-2.9)^2 = 295.39
Now, we can calculate the slope of the line:
Slope (m) = Σ(xy) / Σ(x^2) = (12.98 + 53.58 + 40.33 + 17.25 + 5.61 + 26.26 - 3.77) / 295.39 = 0.37
Finally, we can calculate the y-intercept of the line using the slope and the mean values:
y-intercept (b) = ȳ - (m * x̄) = 10.8 - (0.37 * 17.9) = 10.1
Therefore, the equation of the line of best fit is:
y = 0.37x + 10.1
To determine how many seconds we should expect a player to take to run 40 yards if they can do 22 leg-press repetitions, we can substitute x = 22 into the equation:
y = 0.37(22) + 10.1 = 8.14
Therefore, we should expect a player to take 8.1 seconds to run 40 yards if they can do 22 leg-press repetitions.