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
(2 points)
To find the equation of the line of best fit, we first need to calculate the correlation coefficient and the equation of the line.
Step 1: Calculate the correlation coefficient:
- The correlation coefficient, denoted by r, can be calculated using the following formula:
r = Σ((X - X̄)(Y - Ȳ)) / √(Σ(X - X̄)² * Σ(Y - Ȳ)²)
Where:
- X̄ is the mean of the leg press repetitions.
- Ȳ is the mean of the 40-yard dash times.
Step 2: Calculate the slope, b, of the line of best fit:
b = r * (Sy / Sx)
Where:
- Sx is the standard deviation of the leg press repetitions.
- Sy is the standard deviation of the 40-yard dash times.
Step 3: Calculate the y-intercept, a, of the line of best fit:
a = Ȳ - b * X̄
Where:
- X̄ and Ȳ are the means of the leg press repetitions and 40-yard dash times, respectively.
Step 4: Write the equation of the line of best fit:
Y = a + b * X
Now, let's calculate the equation of the line of best fit:
Mean of leg press repetitions (X̄) = (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 19.1
Mean of 40-yard dash times (Ȳ) = (8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.757
Correlation coefficient (r) = 0.98 (Calculate using the formula above)
Standard deviation of leg press repetitions (Sx) = 8.87
Standard deviation of 40-yard dash times (Sy) = 2.23
Slope (b) = 0.426 (Calculate using the formula above)
Y-intercept (a) = 2.049 (Calculate using the formula above)
Therefore, the equation of the line of best fit is:
Y = 2.049 + 0.426X
Now, to find the expected time it takes for a player to run 40 yards if they can do 22 leg press repetitions:
Y = 2.049 + 0.426 * 22
Y = 11.45 seconds
Therefore, the player can be expected to take approximately 11.5 seconds to run 40 yards.