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 need to calculate the slope and y-intercept.
First, let's calculate the average leg press repetitions and 40-yard dash times:
Average leg press repetitions: (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 18.1
Average 40-yard dash time: (8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.9
Now, let's calculate the sum of the products of the differences between each leg press repetition and the average leg press repetitions, and each 40-yard dash time and the average 40-yard dash time:
Sum of products = (12 - 18.1) * (8.6 - 10.9) + (32 - 18.1) * (14.6 - 10.9) + (7 - 18.1) * (7.1 - 10.9) + (11 - 18.1) * (8.3 - 10.9) + (23 - 18.1) * (11.9 - 10.9) + (28 - 18.1) * (13.4 - 10.9) + (15 - 18.1) * (9.5 - 10.9)
Sum of products = -42.39
Next, let's calculate the sum of the squares of the differences between each leg press repetition and the average leg press repetitions:
Sum of squared differences = (12 - 18.1)^2 + (32 - 18.1)^2 + (7 - 18.1)^2 + (11 - 18.1)^2 + (23 - 18.1)^2 + (28 - 18.1)^2 + (15 - 18.1)^2
Sum of squared differences = 849.7
Now, let's calculate the slope of the line of best fit:
slope = (Sum of products) / (Sum of squared differences)
slope = -42.39 / 849.7
slope ≈ -0.05
Finally, let's calculate the y-intercept of the line of best fit:
y-intercept = average 40-yard dash time - (slope * average leg press repetitions)
y-intercept = 10.9 - (-0.05 * 18.1)
y-intercept ≈ 11.8
Therefore, the equation of the line of best fit is:
y = -0.05x + 11.8
To find how many seconds a player would take to run 40 yards if they can do 22 leg-press repetitions, we can substitute x = 22 into the equation and solve for y:
y = -0.05(22) + 11.8
y ≈ 10.7
Therefore, a player who can do 22 leg press repetitions should be expected to take approximately 10.7 seconds to run 40 yards.
are you sure?
Apologies for the previous incorrect response. Let's recalculate the equation of the line of best fit correctly:
Let's calculate the values needed for the equation of the line of best fit:
n = 7 (number of data points)
Calculate the sum of leg press repetitions and 40-yard dash times:
Sum of leg press repetitions = 12 + 32 + 7 + 11 + 23 + 28 + 15 = 128
Sum of 40-yard dash times = 8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5 = 73.4
Calculate the mean of leg press repetitions and 40-yard dash times:
Mean of leg press repetitions = (Sum of leg press repetitions) / n = 128 / 7 ≈ 18.3
Mean of 40-yard dash times = (Sum of 40-yard dash times) / n = 73.4 / 7 ≈ 10.5
Now, calculate the sum of the products of the differences between each leg press repetition and the mean of leg press repetitions, and each 40-yard dash time and the mean of 40-yard dash times:
Sum of products = (12 - 18.3) * (8.6 - 10.5) + (32 - 18.3) * (14.6 - 10.5) + (7 - 18.3) * (7.1 - 10.5) + (11 - 18.3) * (8.3 - 10.5) + (23 - 18.3) * (11.9 - 10.5) + (28 - 18.3) * (13.4 - 10.5) + (15 - 18.3) * (9.5 - 10.5) ≈ -97.37
Next, calculate the sum of the squared differences between each leg press repetition and the mean of leg press repetitions:
Sum of squared differences = (12 - 18.3)^2 + (32 - 18.3)^2 + (7 - 18.3)^2 + (11 - 18.3)^2 + (23 - 18.3)^2 + (28 - 18.3)^2 + (15 - 18.3)^2 ≈ 1421.89
Calculate the slope of the line of best fit:
slope = (Sum of products) / (Sum of squared differences) ≈ -97.37 / 1421.89 ≈ -0.0685
Lastly, calculate the y-intercept of the line of best fit:
y-intercept = mean of 40-yard dash times - (slope * mean of leg press repetitions) ≈ 10.5 - (-0.0685 * 18.3) ≈ 11.8
Therefore, the equation of the line of best fit is:
y = -0.0685x + 11.8
To find how many seconds a player would take to run 40 yards if they can do 22 leg-press repetitions, we can substitute x = 22 into the equation and solve for y:
y = -0.0685(22) + 11.8
y ≈ 10.03
Therefore, a player who can do 22 leg press repetitions should be expected to take approximately 10.03 seconds to run 40 yards.