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
To find the equation of the line of best fit, we can use the least squares regression method. The equation of a straight line is given by y = mx + b, where m is the slope of the line and b is the y-intercept.
First, let's calculate the slope (m):
m = Σ((x - x̄)(y - ȳ)) / Σ(x - x̄)²
Where:
x̄ = average number of leg-press repetitions
ȳ = average 40-yard dash time
(x, y) = data points
Using the provided data:
Player 1: (20, 5.2)
Player 2: (24, 5.1)
Player 3: (22, 5.3)
Player 4: (18, 5.5)
Player 5: (23, 5.0)
Player 6: (21, 5.4)
Player 7: (19, 5.6)
Calculating the averages:
x̄ = (20 + 24 + 22 + 18 + 23 + 21 + 19) / 7 = 21.4
ȳ = (5.2 + 5.1 + 5.3 + 5.5 + 5.0 + 5.4 + 5.6) / 7 = 5.3
Now plug the values into the slope formula:
m = ((20 - 21.4)(5.2 - 5.3) + (24 - 21.4)(5.1 - 5.3) + (22 - 21.4)(5.3 - 5.3) + (18 - 21.4)(5.5 - 5.3) + (23 - 21.4)(5.0 - 5.3) + (21 - 21.4)(5.4 - 5.3) + (19 - 21.4)(5.6 - 5.3)) / ((20 - 21.4)² + (24 - 21.4)² + (22 - 21.4)² + (18 - 21.4)² + (23 - 21.4)² + (21 - 21.4)² + (19 - 21.4)²)
m = (-1.4 * -0.1 + 2.6 * -0.2 + 0.6 * 0 + -3.4 * 0.2 + 1.6 * -0.3 + -0.4 * 0.1 + -2.4 * 0.3) / (1.4² + 2.6² + 0.6² + 3.4² + 1.6² + 0.4² + 2.4²)
m = (-0.14 - 0.52 + 0 + 0.68 - 0.48 - 0.04 - 0.72) / (1.96 + 6.76 + 0.36 + 11.56 + 2.56 + 0.16 + 5.76)
m = -0.12 / 28.12
m = -0.0043
Now, calculate the y-intercept (b):
b = ȳ - (m * x̄)
b = 5.3 - (-0.0043 * 21.4)
b = 5.3 + 0.092
b = 5.392
Therefore, the equation of the line of best fit is:
y = -0.0043x + 5.392
To find out how many seconds a player should expect to take to run 40 yards if they can do 22 leg-press repetitions:
y = -0.0043(22) + 5.392
y = 5.298
So, the player should expect to take approximately 5.3 seconds to run 40 yards.