A college football coach wants to know if the is a correlation between his players' 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 tables 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?
Leg Press (reps)|40-yard Dash (s)
15 |5.2
18 |6.3
8 |6.8
30 |8.2
26 |8.0
12 |5.3
21 |
I need help creating an equation
40=30/8(26+8)
To create an equation for the line of best fit, we can use linear regression analysis. The equation of a line is typically written in the form of y = mx + b, where y represents the dependent variable (40-yard dash time in this case), x represents the independent variable (number of leg press reps), m represents the slope of the line, and b represents the y-intercept.
To find the equation of the line of best fit, we need to calculate the slope (m) and the y-intercept (b) using the given data points.
I will calculate the slope and y-intercept step-by-step:
Step 1: Compute the means of the independent variable (leg press reps) and dependent variable (40-yard dash time) and denote them as x̄ and ȳ, respectively.
x̄ (mean of leg press reps) = (15 + 18 + 8 + 30 + 26 + 12 + 21) / 7 = 18
ȳ (mean of 40-yard dash times) = (5.2 + 6.3 + 6.8 + 8.2 + 8.0 + 5.3) / 6 = 6.37
Step 2: Calculate the deviations from the mean for both variables (x and y).
Deviation from x̄ (Dx) = leg press reps - x̄
Deviation from ȳ (Dy) = 40-yard dash time - ȳ
Using the data provided, we have:
Dx = (15 - 18, 18 - 18, 8 - 18, 30 - 18, 26 - 18, 12 - 18, 21 - 18)
= (-3, 0, -10, 12, 8, -6, 3)
Dy = (5.2 - 6.37, 6.3 - 6.37, 6.8 - 6.37, 8.2 - 6.37, 8.0 - 6.37, 5.3 - 6.37)
= (-1.17, -0.07, 0.43, 1.83, 1.63, -1.07)
Step 3: Calculate the sum of products of the deviations of x and y (DPxy) and the sum of squared deviations of x (DSx).
DPxy = (Dx1 * Dy1) + (Dx2 * Dy2) + ... + (Dxn * Dyn)
= (-3 * -1.17) + (0 * -0.07) + (-10 * 0.43) + (12 * 1.83) + (8 * 1.63) + (-6 * -1.07) + (3 * 0.00)
= 3.51 + 0 + -4.30 + 21.96 + 13.04 + 6.42 + 0
= 40.63
DSx = (Dx1 * Dx1) + (Dx2 * Dx2) + ... + (Dxn * Dxn)
= (-3 * -3) + (0 * 0) + (-10 * -10) + (12 * 12) + (8 * 8) + (-6 * -6) + (3 * 3)
= 9 + 0 + 100 + 144 + 64 + 36 + 9
= 362
Step 4: Calculate the slope (m).
m = DPxy / DSx = 40.63 / 362 = 0.112
Step 5: Calculate the y-intercept (b).
b = ȳ - m * x̄ = 6.37 - 0.112 * 18 = 6.37 - 2.016 = 4.354
Therefore, the equation of the line of best fit is:
y = 0.112x + 4.354
To find the expected 40-yard dash time for a player who can do 22 leg-press repetitions, substitute x = 22 into the equation:
y = 0.112 * 22 + 4.354
y = 2.464 + 4.354
y = 6.818
Therefore, if a player can do 22 leg-press repetitions, the coach should expect the player to take approximately 6.818 seconds to run 40 yards.