A college football coach wants to know if the 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 tables 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?
Leg Press (reps)|40-yard Dash (s)
15 |5.2
18 |6.3
8 |6.8
30 |8.2
26 |8.0
12 |5.3
21 |
I need help creating an equation
40=30/8(26+8)
To create an equation of the line of best fit, we will use linear regression analysis. This analysis will help us determine the relationship between the number of leg press repetitions and the time it takes to sprint 40 yards.
Step 1: Arrange the data in ascending order based on the number of leg press repetitions.
Leg Press (reps): 8, 12, 15, 18, 21, 26, 30
40-yard Dash (s): 6.8, 5.3, 5.2, 6.3, ?, 8.0, 8.2
Step 2: Find the averages of both the leg press repetitions and 40-yard dash times.
Average leg press repetitions (x̄)= (8+12+15+18+21+26+30)/7 = 18.71
Average 40-yard dash time (ȳ) = (6.8+5.3+5.2+6.3+8.0+8.2)/6 ≈ 6.47
Step 3: Calculate the sum of the products of the deviation of each value from its average for both variables.
Σ(x-x̄)(y-ȳ) = (8-18.71)(6.8-6.47) + (12-18.71)(5.3-6.47) + (15-18.71)(5.2-6.47) + (18-18.71)(6.3-6.47) + (21-18.71)(?-6.47) + (26-18.71)(8.0-6.47) + (30-18.71)(8.2-6.47)
Step 4: Calculate the sum of squares of the deviation for the independent variable.
Σ(x-x̄)² = (8-18.71)² + (12-18.71)² + (15-18.71)² + (18-18.71)² + (21-18.71)² + (26-18.71)² + (30-18.71)²
Step 5: Calculate the slope of the line of best fit using the following formula:
slope (b) = Σ(x-x̄)(y-ȳ) / Σ(x-x̄)²
Step 6: Calculate the y-intercept of the line of best fit using the following formula:
y-intercept (a) = ȳ - (b*x̄)
Plugging the values into the formulas, we get:
Σ(x-x̄)(y-ȳ) = (-10.71)(0.33) + (-6.71)(-1.17) + (-3.71)(-1.27) + (-0.71)(0.83) + (2.29)(?-1.17) + (7.29)(1.53) + (11.29)(1.73) ≈ 57.86
Σ(x-x̄)² = (-10.71)² + (-6.71)² + (-3.71)² + (-0.71)² + (2.29)² + (7.29)² + (11.29)² ≈ 589.86
slope (b) = 57.86 / 589.86 ≈ 0.098
y-intercept (a) = 6.47 - (0.098 * 18.71) ≈ 4.625
Therefore, the equation of the line of best fit is:
40-yard Dash (s) ≈ 4.625 + 0.098 * Leg Press (reps)
To find out how many seconds a player should expect to take to run 40 yards if that player can do 22 leg press repetitions, plug in Leg Press (reps) = 22 into the equation:
40-yard Dash (s) ≈ 4.625 + 0.098 * 22
40-yard Dash (s) ≈ 4.625 + 2.156
40-yard Dash (s) ≈ 6.781
Therefore, a player who can do 22 leg press repetitions is expected to take approximately 6.781 seconds to run 40 yards.