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
I apologize for the mistake.
Let's recalculate the slope and y-intercept.
First, let's calculate the average values for both leg press repetitions and 40-yard dash times:
Average leg press repetitions: (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 18
Average 40-yard dash time: (8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.4
Next, let's calculate the deviations from the averages for both variables:
For leg press repetitions:
12 - 18 = -6
32 - 18 = 14
7 - 18 = -11
11 - 18 = -7
23 - 18 = 5
28 - 18 = 10
15 - 18 = -3
For 40-yard dash times:
8.6 - 10.4 = -1.8
14.6 - 10.4 = 4.2
7.1 - 10.4 = -3.3
8.3 - 10.4 = -2.1
11.9 - 10.4 = 1.5
13.4 - 10.4 = 3.0
9.5 - 10.4 = -0.9
Now, let's calculate the sum of the products of the deviations:
-6 * -1.8 + 14 * 4.2 + (-11) * (-3.3) + (-7) * (-2.1) + 5 * 1.5 + 10 * 3.0 + (-3) * (-0.9) = 132.9
Next, let's calculate the sum of the squares of the deviations for both variables:
Sum of the squares of the leg press deviations = (-6)^2 + (14)^2 + (-11)^2 + (-7)^2 + (5)^2 + (10)^2 + (-3)^2 = 410
Sum of the squares of the 40-yard dash time deviations = (-1.8)^2 + (4.2)^2 + (-3.3)^2 + (-2.1)^2 + (1.5)^2 + (3.0)^2 + (-0.9)^2 = 34.38
Now, let's calculate the slope of the line:
slope = sum of products of deviations / sum of squares of leg press deviations = 132.9 / 410 = 0.324
Now we can calculate the y-intercept:
y-intercept = average 40-yard dash time - slope * average leg press repetitions = 10.4 - 0.324 * 18 = 4.8
Therefore, the equation of the line of best fit is:
40-yard dash time = 0.324 * leg press repetitions + 4.8
To find how many seconds a player should take to run 40 yards if they can do 22 leg press repetitions:
40-yard dash time = 0.324 * 22 + 4.8 = 11.3 seconds.
Therefore, the player should expect to take approximately 11.3 seconds to run 40 yards.
the one that says i agologize for the mistake is Correct
To find the equation of the line of best fit, we need to calculate the slope and y-intercept.
First, let's calculate the average values for both leg press repetitions and 40-yard dash times:
Average leg press repetitions: (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 19.9
Average 40-yard dash time: (8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.9
Next, let's calculate the deviations from the averages for both variables:
For leg press repetitions:
12 - 19.9 = -7.9
32 - 19.9 = 12.1
7 - 19.9 = -12.9
11 - 19.9 = -8.9
23 - 19.9 = 3.1
28 - 19.9 = 8.1
15 - 19.9 = -4.9
For 40-yard dash times:
8.6 - 10.9 = -2.3
14.6 - 10.9 = 3.7
7.1 - 10.9 = -3.8
8.3 - 10.9 = -2.6
11.9 - 10.9 = 1.0
13.4 - 10.9 = 2.5
9.5 - 10.9 = -1.4
Now, let's calculate the sum of the products of the deviations:
-7.9 * -2.3 + 12.1 * 3.7 + (-12.9) * (-3.8) + (-8.9) * (-2.6) + 3.1 * 1.0 + 8.1 * 2.5 + (-4.9) * (-1.4) = 77.68
Next, let's calculate the sum of the squares of the deviations for both variables:
Sum of the squares of the leg press deviations = (-7.9)^2 + (12.1)^2 + (-12.9)^2 + (-8.9)^2 + (3.1)^2 + (8.1)^2 + (-4.9)^2 = 1021.78
Sum of the squares of the 40-yard dash time deviations = (-2.3)^2 + (3.7)^2 + (-3.8)^2 + (-2.6)^2 + (1.0)^2 + (2.5)^2 + (-1.4)^2 = 54.57
Now, let's calculate the slope of the line:
slope = sum of products of deviations / sum of squares of leg press deviations = 77.68 / 1021.78 = 0.076
Now we can calculate the y-intercept:
y-intercept = average 40-yard dash time - slope * average leg press repetitions = 10.9 - 0.076 * 19.9 = 9.4
Therefore, the equation of the line of best fit is:
40-yard dash time = 0.076 * leg press repetitions + 9.4
To find how many seconds a player should take to run 40 yards if they can do 22 leg press repetitions:
40-yard dash time = 0.076 * 22 + 9.4 = 11.1 seconds.
Therefore, the player should expect to take approximately 11.1 seconds to run 40 yards.
are you sure
I apologize for the confusion. The correct equation of the line of best fit is:
40-yard dash time = 0.076 * leg press repetitions + 9.4
And for a player who can do 22 leg press repetitions, they should expect to take approximately 11.1 seconds to run 40 yards.