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.
To find the equation of the line of best fit, we can use the method of linear regression. This involves finding the equation of a straight line that best fits the given data points.
First, let's arrange the data in a table:
| Leg-Press Repetitions (x) | 40-Yard Dash Time (y) |
|--------------------------|----------------------|
| 12 | 6.9 |
| 14 | 6.5 |
| 16 | 6.1 |
| 18 | 5.8 |
| 20 | 5.4 |
| 20 | 5.2 |
| 26 | 4.9 |
Using this data, we can calculate the slope and y-intercept of the line of best fit.
First, let's calculate the mean of the x-values (leg-press repetitions) and the mean of the y-values (40-yard dash times):
mean(x) = (12 + 14 + 16 + 18 + 20 + 20 + 26)/7 = 18
mean(y) = (6.9 + 6.5 + 6.1 + 5.8 + 5.4 + 5.2 + 4.9)/7 = 5.97
Next, let's calculate the sum of the products of (x - mean(x)) and (y - mean(y)):
Σ((x - mean(x))(y - mean(y))) = ((12-18)(6.9-5.97)) + ((14-18)(6.5-5.97)) + ((16-18)(6.1-5.97)) + ((18-18)(5.8-5.97)) + ((20-18)(5.4-5.97)) + ((20-18)(5.2-5.97)) + ((26-18)(4.9-5.97))
= (-6)(0.93) + (-4)(0.53) + (-2)(0.13) + (0)(-0.17) + (2)(-0.57) + (2)(-0.77) + (8)(-1.07)
= -5.58 - 2.12 - 0.26 + 0 + (-1.14) + (-1.54) + (-8.56)
= -19.92
Next, let's calculate the sum of the squares of (x - mean(x)):
Σ((x - mean(x))^2) = ((12-18)^2) + ((14-18)^2) + ((16-18)^2) + ((18-18)^2) + ((20-18)^2) + ((20-18)^2) + ((26-18)^2)
= 36 + 16 + 4 + 0 + 4 + 4 + 64
= 128
Using these values, we can now calculate the slope of the line of best fit:
slope = Σ((x - mean(x))(y - mean(y)))/Σ((x - mean(x))^2) = -19.92/128 = -0.155625
Now that we have the slope, we can use the point-slope form of a linear equation to find the y-intercept:
y - mean(y) = slope(x - mean(x))
y - 5.97 = -0.155625(x - 18)
y - 5.97 = -0.155625x + 2.809375
y = -0.155625x + 8.779375
Therefore, the equation of the line of best fit is y = -0.155625x + 8.779375.
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 = -0.155625(22) + 8.779375
y = -3.424375 + 8.779375
y ≈ 5.355
Therefore, a player who can do 22 leg-press repetitions would be expected to take approximately 5.4 seconds to run 40 yards.