A college football coach wants to know if there is a correlation between his players’ 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)
15
18
8
30
26
12
21
40-yard Dash (s)
5.2
6.3
6.8
8.2
8.0
5.3
5.9
To find the equation of the line of best fit and predict the 40-yard dash time using leg press repetitions, we need to perform linear regression analysis.
1. First, we calculate the mean of the leg press repetitions and the mean of the 40-yard dash times.
Leg Press (reps) Mean:
(15 + 18 + 8 + 30 + 26 + 12 + 21) / 7 = 130 / 7 = 18.6 (rounded to one decimal place)
40-yard Dash (s) Mean:
(5.2 + 6.3 + 6.8 + 8.2 + 8.0 + 5.3 + 5.9) / 7 = 45.7 / 7 = 6.53 (rounded to two decimal places)
2. Next, we calculate the sum of the products and the sum of the squares.
Sum of the Products:
(15 - 18.6)*(5.2 - 6.53) + (18 - 18.6)*(6.3 - 6.53) + (8 - 18.6)*(6.8 - 6.53)
+ (30 - 18.6)*(8.2 - 6.53) + (26 - 18.6)*(8.0 - 6.53) + (12 - 18.6)*(5.3 - 6.53)
+ (21 - 18.6)*(5.9 - 6.53) = -108
Sum of the Squares of Leg Press (reps):
(15 - 18.6)^2 + (18 - 18.6)^2 + (8 - 18.6)^2 + (30 - 18.6)^2 + (26 - 18.6)^2 + (12 - 18.6)^2 + (21 - 18.6)^2 = 735.8
Sum of the Squares of 40-yard Dash (s):
(5.2 - 6.53)^2 + (6.3 - 6.53)^2 + (6.8 - 6.53)^2 + (8.2 - 6.53)^2 + (8.0 - 6.53)^2 + (5.3 - 6.53)^2 + (5.9 - 6.53)^2 = 8.63
3. We calculate the slope and intercept of the line of best fit.
Slope:
slope = sum of the products / sum of the squares of leg press reps
slope = -108 / 735.8 ≈ -0.1470 (rounded to four decimal places)
Intercept:
intercept = mean of 40-yard dash times - (slope * mean of leg press reps)
intercept = 6.53 - (-0.1470 * 18.6)
intercept ≈ 9.72 (rounded to two decimal places)
4. Finally, we write the equation of the line of best fit in the form y = mx + b, where y is the 40-yard dash time and x is the leg press reps.
Equation of the Line of Best Fit:
y = -0.1470x + 9.72
To predict the 40-yard dash time for a player who can do 22 leg-press repetitions, substitute x = 22 into the equation:
y = -0.1470 * 22 + 9.72 ≈ 6.15 seconds (rounded to two decimal places)
Therefore, the coach should expect a player who can do 22 leg-press repetitions to take approximately 6.15 seconds to run 40 yards.
I assume you have learned how to figure the line of best fit.
Once you have that line, y=mx+b, then just plug in x=22
You can use google to find best-fit line calculators to check your answer.