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 |5.9
Please help And show all the steps you used :)
To find the equation of the line of best fit, you can use linear regression. Linear regression helps determine the relationship between two variables and finds the line that best represents that relationship. In this case, the independent variable is the leg press repetitions, and the dependent variable is the 40-yard dash time.
Step 1: Calculate the mean for both the leg press repetitions and the 40-yard dash time.
Mean of leg press repetitions (x̄):
(15 + 18 + 8 + 30 + 26 + 12 + 21) / 7 = 20
Mean of 40-yard dash time (ȳ):
(5.2 + 6.3 + 6.8 + 8.2 + 8.0 + 5.3 + 5.9) / 7 = 6.5
Step 2: Calculate the deviations from the mean for both variables.
Deviation of leg press repetitions (x):
15 - 20 = -5
18 - 20 = -2
8 - 20 = -12
30 - 20 = 10
26 - 20 = 6
12 - 20 = -8
21 - 20 = 1
Deviation of 40-yard dash time (y):
5.2 - 6.5 = -1.3
6.3 - 6.5 = -0.2
6.8 - 6.5 = 0.3
8.2 - 6.5 = 1.7
8.0 - 6.5 = 1.5
5.3 - 6.5 = -1.2
5.9 - 6.5 = -0.6
Step 3: Calculate the products of the deviations for each data point.
Product of deviations (xy):
(-5)(-1.3) = 6.5
(-2)(-0.2) = 0.4
(-12)(0.3) = -3.6
(10)(1.7) = 17
(6)(1.5) = 9
(-8)(-1.2) = 9.6
(1)(-0.6) = -0.6
Step 4: Calculate the squared deviations for each variable.
Squared deviation of leg press repetitions (x^2):
(-5)^2 = 25
(-2)^2 = 4
(-12)^2 = 144
(10)^2 = 100
(6)^2 = 36
(-8)^2 = 64
(1)^2 = 1
Squared deviation of 40-yard dash time (y^2):
(-1.3)^2 = 1.69
(-0.2)^2 = 0.04
(0.3)^2 = 0.09
(1.7)^2 = 2.89
(1.5)^2 = 2.25
(-1.2)^2 = 1.44
(-0.6)^2 = 0.36
Step 5: Calculate the sum of all the values calculated in steps 2, 3, 4.
Sum of x, y, xy, x^2, and y^2:
Σx = -5 + (-2) + (-12) + 10 + 6 + (-8) + 1 = -10
Σy = -1.3 + (-0.2) + 0.3 + 1.7 + 1.5 + (-1.2) + (-0.6) = 0.2
Σxy = 6.5 + 0.4 + (-3.6) + 17 + 9 + 9.6 + (-0.6) = 38.3
Σx^2 = 25 + 4 + 144 + 100 + 36 + 64 + 1 = 374
Σy^2 = 1.69 + 0.04 + 0.09 + 2.89 + 2.25 + 1.44 + 0.36 = 9.16
Step 6: Calculate the slope (m) using the formula:
m = (n * Σxy - Σx * Σy) / (n * Σx^2 - Σx^2)
where n is the number of data points.
m = (7 * 38.3 - (-10) * 0.2) / (7 * 374 - 10^2)
m = 181.9 / 2248
m ≈ 0.0809 (rounded to four decimal places)
Step 7: Calculate the y-intercept (b) using the formula:
b = ȳ - m * x̄
b = 6.5 - 0.0809 * 20
b = 6.5 - 1.618
b ≈ 4.882 (rounded to three decimal places)
Step 8: Write the equation of the line of best fit using the slope and y-intercept from steps 6 and 7, respectively.
y = mx + b
y = 0.0809x + 4.882
The equation of the line of best fit is y = 0.0809x + 4.882.
Step 9: Determine the predicted 40-yard dash time for a leg press repetition of 22.
y = 0.0809 * 22 + 4.882
y ≈ 6.66 (rounded to two decimal places)
Therefore, if a player can do 22 leg press repetitions, the coach can expect them to take approximately 6.66 seconds to run 40 yards.