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.
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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 will use the least squares method to find the slope and intercept of the line that best fits the data.
First, we calculate the means of both sets of data:
Mean of leg press repetitions = (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 16.1
Mean of 40-yard dash times = (8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.8
Next, we calculate the deviations from the mean for both sets of data:
Leg press deviation = [12-16.1, 32-16.1, 7-16.1, 11-16.1, 23-16.1, 28-16.1, 15-16.1]
40-yard dash deviation = [8.6-10.8, 14.6-10.8, 7.1-10.8, 8.3-10.8, 11.9-10.8, 13.4-10.8, 9.5-10.8]
Now, we calculate the sum of the products of the deviations:
Sum of (Leg press deviation * 40-yard dash deviation) = (12-16.1)*(8.6-10.8) + (32-16.1)*(14.6-10.8) + (7-16.1)*(7.1-10.8) + (11-16.1)*(8.3-10.8) + (23-16.1)*(11.9-10.8) + (28-16.1)*(13.4-10.8) + (15-16.1)*(9.5-10.8)
Now, we calculate the sum of the squares of the leg press deviations:
Sum of (Leg press deviation)^2 = (12-16.1)^2 + (32-16.1)^2 + (7-16.1)^2 + (11-16.1)^2 + (23-16.1)^2 + (28-16.1)^2 + (15-16.1)^2
The slope of the line of best fit is given by:
slope = Sum of (Leg press deviation * 40-yard dash deviation) / Sum of (Leg press deviation)^2
The y-intercept of the line of best fit is given by:
y-intercept = Mean of 40-yard dash times - (slope * Mean of leg press repetitions)
Once the slope and y-intercept are calculated, the equation of the line of best fit is in the form:
y = slope * x + y-intercept
Using the values calculated for the slope and y-intercept, the equation of the line of best fit is determined. Once the equation is found, we can substitute x=22 (number of leg-press repetitions) to determine the expected 40-yard dash time. Let me calculate these values.