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
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.
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.
To find the equation of the line of best fit, we can use linear regression analysis. This analysis will provide us with the equation of a straight line that best represents the relationship between the leg press repetitions and the 40-yard dash time.
Step 1: Calculate the mean (average) of both the leg press repetitions and the 40-yard dash times.
Mean of leg press reps = (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 = 10.657142857142857 (approximately 10.7)
Step 2: Calculate the differences from the mean for both leg press reps and 40-yard dash times.
For leg press reps:
Differences from the mean = leg press reps - mean of leg press reps
For 40-yard dash times:
Differences from the mean = 40-yard dash times - mean of 40-yard dash times
Leg press reps differences from the mean: 12 - 17 = -5, 32 - 17 = 15, 7 - 17 = -10, 11 - 17 = -6, 23 - 17 = 6, 28 - 17 = 11, 15 - 17 = -2
40-yard dash times differences from the mean: 8.6 - 10.7 = -2.1, 14.6 - 10.7 = 3.9, 7.1 - 10.7 = -3.6, 8.3 - 10.7 = -2.4, 11.9 - 10.7 = 1.2, 13.4 - 10.7 = 2.7, 9.5 - 10.7 = -1.2
Step 3: Calculate the product of the differences from the mean for each data point.
Product = difference from the mean of leg press reps * difference from the mean of 40-yard dash times
-5 * -2.1 = 10.5, 15 * 3.9 = 58.5, -10 * -3.6 = 36, -6 * -2.4 = 14.4, 6 * 1.2 = 7.2, 11 * 2.7 = 29.7, -2 * -1.2 = 2.4
Step 4: Calculate the square of the differences from the mean for each variable.
Square of leg press reps differences from the mean: (-5)^2 = 25, (15)^2 = 225, (-10)^2 = 100, (-6)^2 = 36, (6)^2 = 36, (11)^2 = 121, (-2)^2 = 4
Square of 40-yard dash times differences from the mean: (-2.1)^2 = 4.41, (3.9)^2 = 15.21, (-3.6)^2 = 12.96, (-2.4)^2 = 5.76, (1.2)^2 = 1.44, (2.7)^2 = 7.29, (-1.2)^2 = 1.44
Step 5: Sum up the product and the squared differences from the mean.
Sum of product = 10.5 + 58.5 + 36 + 14.4 + 7.2 + 29.7 + 2.4 = 158.7
Sum of square of leg press reps differences = 25 + 225 + 100 + 36 + 36 + 121 + 4 = 547
Sum of square of 40-yard dash times differences = 4.41 + 15.21 + 12.96 + 5.76 + 1.44 + 7.29 + 1.44 = 48.51
Step 6: Calculate the slope (b) of the line of best fit.
Slope (b) = Sum of product / Sum of square of leg press reps differences
b = 158.7 / 547 ≈ 0.290 (rounded to three decimal places)
Step 7: Calculate the y-intercept (a) of the line of best fit.
y-intercept (a) = mean of 40-yard dash times - (slope * mean of leg press reps)
a = 10.7 - (0.290 * 17) = 10.706 (rounded to three decimal places)
Step 8: Write the equation of the line of best fit.
The equation of the line of best fit is y = 0.290x + 10.706.
To find how many seconds a player should take to run 40 yards if they can do 22 leg press repetitions, substitute x = 22 into the equation.
y = 0.290(22) + 10.706
y ≈ 16.98 (rounded to two decimal places)
Therefore, the player should be expected to take approximately 17 seconds to run 40 yards if they can do 22 leg press repetitions.