To find the equation of the line of best fit, we can use linear regression to calculate the equation in slope-intercept form (y = mx + b).
First, let's calculate the mean of the leg press repetitions (x) and the mean of the 40-yard dash times (y).
Mean of leg press repetitions:
(12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 18.1
Mean of 40-yard dash times:
(8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.77
Next, let's calculate the sum of the products of each leg press repetition and 40-yard dash time, as well as the sum of the squares of the leg press repetitions.
Sum of products:
(12 * 8.6) + (32 * 14.6) + (7 * 7.1) + (11 * 8.3) + (23 * 11.9) + (28 * 13.4) + (15 * 9.5) = 2826.9
Sum of squares of leg press repetitions:
(12^2) + (32^2) + (7^2) + (11^2) + (23^2) + (28^2) + (15^2) = 3824
Next, let's calculate the sum of the squares of the 40-yard dash times.
Sum of squares of 40-yard dash times:
(8.6^2) + (14.6^2) + (7.1^2) + (8.3^2) + (11.9^2) + (13.4^2) + (9.5^2) = 818.42
Next, let's calculate the sum of the products of each leg press repetition and 40-yard dash time minus n times the mean of leg press repetitions times the mean of 40-yard dash times.
Sum of products minus n times mean of leg press repetitions times mean of 40-yard dash times:
2826.9 - (7 * 18.1 * 10.77) = 1053.2
Next, let's calculate the sum of squares of leg press repetitions minus n times the mean of leg press repetitions squared.
Sum of squares of leg press repetitions minus n times mean of leg press repetitions squared:
3824 - (7 * 18.1^2) = 2359.96
Now, let's calculate the slope (m) of the line of best fit:
m = (Sum of products minus n times mean of leg press repetitions times mean of 40-yard dash times) / (Sum of squares of leg press repetitions minus n times mean of leg press repetitions squared)
m = 1053.2 / 2359.96 ≈ 0.446
Next, let's calculate the y-intercept (b) of the line of best fit:
b = mean of 40-yard dash times - (slope * mean of leg press repetitions)
b = 10.77 - (0.446 * 18.1) ≈ 3.827
Therefore, the equation of the line of best fit is:
y = 0.446x + 3.827
To find how many seconds a player should expect to take to run 40 yards if they can do 22 leg press repetitions, we substitute x = 22 into the equation:
y = 0.446(22) + 3.827
y ≈ 12.6
Therefore, the player should expect to take approximately 12.6 seconds to run 40 yards.
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 (s)
8.6
14.6
7.1
8.3
11.9
13.4
9.5
3 answers
bot it takes 11.9 seconds to run the 40 yard dash so what is the equation of line best fit
Apologies for the error. Let's recalculate the equation of the line of best fit.
Given the data:
Leg Press (reps): 12, 32, 7, 11, 23, 28, 15
40-yard Dash (s): 8.6, 14.6, 7.1, 8.3, 11.9, 13.4, 9.5
Using linear regression, we can calculate the equation in slope-intercept form (y = mx + b).
First, let's calculate the mean of the leg press repetitions (x) and the mean of the 40-yard dash times (y).
Mean of leg press repetitions:
(12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 17.14
Mean of 40-yard dash times:
(8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.3
Next, let's calculate the sum of the products of each leg press repetition and 40-yard dash time, as well as the sum of the squares of the leg press repetitions.
Sum of products:
(12 * 8.6) + (32 * 14.6) + (7 * 7.1) + (11 * 8.3) + (23 * 11.9) + (28 * 13.4) + (15 * 9.5) = 2711.33
Sum of squares of leg press repetitions:
(12^2) + (32^2) + (7^2) + (11^2) + (23^2) + (28^2) + (15^2) = 3388
Next, let's calculate the sum of the squares of the 40-yard dash times.
Sum of squares of 40-yard dash times:
(8.6^2) + (14.6^2) + (7.1^2) + (8.3^2) + (11.9^2) + (13.4^2) + (9.5^2) = 729.96
Next, let's calculate the sum of the products of each leg press repetition and 40-yard dash time minus n times the mean of leg press repetitions times the mean of 40-yard dash times.
Sum of products minus n times mean of leg press repetitions times mean of 40-yard dash times:
2711.33 - (7 * 17.14 * 10.3) = 200.79
Next, let's calculate the sum of squares of leg press repetitions minus n times the mean of leg press repetitions squared.
Sum of squares of leg press repetitions minus n times mean of leg press repetitions squared:
3388 - (7 * 17.14^2) = 1654.04
Now, let's calculate the slope (m) of the line of best fit:
m = (Sum of products minus n times mean of leg press repetitions times mean of 40-yard dash times) / (Sum of squares of leg press repetitions minus n times mean of leg press repetitions squared)
m = 200.79 / 1654.04 ≈ 0.121
Next, let's calculate the y-intercept (b) of the line of best fit:
b = mean of 40-yard dash times - (slope * mean of leg press repetitions)
b = 10.3 - (0.121 * 17.14) ≈ 7.34
Therefore, the equation of the line of best fit is:
y = 0.121x + 7.34
To find the expected time for a player who can do 22 leg press repetitions, we substitute x = 22 into the equation:
y = 0.121(22) + 7.34
y ≈ 9.67
Therefore, the player should be expected to take approximately 9.7 seconds to run 40 yards.
Given the data:
Leg Press (reps): 12, 32, 7, 11, 23, 28, 15
40-yard Dash (s): 8.6, 14.6, 7.1, 8.3, 11.9, 13.4, 9.5
Using linear regression, we can calculate the equation in slope-intercept form (y = mx + b).
First, let's calculate the mean of the leg press repetitions (x) and the mean of the 40-yard dash times (y).
Mean of leg press repetitions:
(12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 17.14
Mean of 40-yard dash times:
(8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.3
Next, let's calculate the sum of the products of each leg press repetition and 40-yard dash time, as well as the sum of the squares of the leg press repetitions.
Sum of products:
(12 * 8.6) + (32 * 14.6) + (7 * 7.1) + (11 * 8.3) + (23 * 11.9) + (28 * 13.4) + (15 * 9.5) = 2711.33
Sum of squares of leg press repetitions:
(12^2) + (32^2) + (7^2) + (11^2) + (23^2) + (28^2) + (15^2) = 3388
Next, let's calculate the sum of the squares of the 40-yard dash times.
Sum of squares of 40-yard dash times:
(8.6^2) + (14.6^2) + (7.1^2) + (8.3^2) + (11.9^2) + (13.4^2) + (9.5^2) = 729.96
Next, let's calculate the sum of the products of each leg press repetition and 40-yard dash time minus n times the mean of leg press repetitions times the mean of 40-yard dash times.
Sum of products minus n times mean of leg press repetitions times mean of 40-yard dash times:
2711.33 - (7 * 17.14 * 10.3) = 200.79
Next, let's calculate the sum of squares of leg press repetitions minus n times the mean of leg press repetitions squared.
Sum of squares of leg press repetitions minus n times mean of leg press repetitions squared:
3388 - (7 * 17.14^2) = 1654.04
Now, let's calculate the slope (m) of the line of best fit:
m = (Sum of products minus n times mean of leg press repetitions times mean of 40-yard dash times) / (Sum of squares of leg press repetitions minus n times mean of leg press repetitions squared)
m = 200.79 / 1654.04 ≈ 0.121
Next, let's calculate the y-intercept (b) of the line of best fit:
b = mean of 40-yard dash times - (slope * mean of leg press repetitions)
b = 10.3 - (0.121 * 17.14) ≈ 7.34
Therefore, the equation of the line of best fit is:
y = 0.121x + 7.34
To find the expected time for a player who can do 22 leg press repetitions, we substitute x = 22 into the equation:
y = 0.121(22) + 7.34
y ≈ 9.67
Therefore, the player should be expected to take approximately 9.7 seconds to run 40 yards.