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 linear regression analysis to determine the slope and y-intercept of the line.
First, let's calculate the mean of the leg press repetitions and the 40-yard dash times:
Mean of leg press repetitions: (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 17
Mean of 40-yard dash times: (8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.5
Next, we need to calculate the sum of the products of each player's leg press repetitions and 40-yard dash times, as well as the sum of the squares of the leg press repetitions:
Sum of (leg press repetitions * 40-yard dash times):
(12 * 8.6) + (32 * 14.6) + (7 * 7.1) + (11 * 8.3) + (23 * 11.9) + (28 * 13.4) + (15 * 9.5) = 1965.39
Sum of (leg press repetitions squared):
12^2 + 32^2 + 7^2 + 11^2 + 23^2 + 28^2 + 15^2 = 2408
Next, let's calculate the sum of the squared leg press repetitions:
Sum of leg press repetitions: (12 + 32 + 7 + 11 + 23 + 28 + 15) = 128
Now, let's use the formulas for slope and y-intercept:
Slope (m) = (n * Σ(xy) - Σx * Σy) / (n * Σ(x^2) - (Σx)^2)
y-intercept (b) = (Σy - m * Σx) / n
Substituting in the values we calculated earlier:
m = (7 * 1965.39 - 128 * 10.5) / (7 * 2408 - (128)^2)
= (13757.73 - 1344) / (16856 - 16384)
= 12413.73 / 472
≈ 26.3
b = (10.5 - 26.3 * 17) / 7
= (10.5 - 444.1) / 7
≈ -61.3
Therefore, the equation of the line of best fit is:
y = 26.3x - 61.3
To find out how many seconds should a player expect to take to run 40 yards if that player can do 22 leg-press repetitions, we can substitute x = 22 into the equation:
y = 26.3 * 22 - 61.3
≈ 570.6 - 61.3
≈ 509.3
Therefore, the player should expect to take approximately 509.3 seconds to run 40 yards.