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 need to determine the slope and y-intercept in the form of y = mx + b, where y represents the 40-yard dash time and x represents the number of leg-press repetitions.
First, let's find the slope:
Using the formula for the slope:
m = (Σ(xy) - (Σx)(Σy) / n(Σx^2) - (Σx)^2)
Using the given values in the table:
x = [35, 32, 38, 33, 31, 36, 30] (leg-press repetitions)
y = [5.1, 5.3, 5.2, 5.5, 5.4, 5.3, 5.7] (40-yard dash time in seconds)
n = 7 (number of data points)
Σ(xy) = (35*5.1) + (32*5.3) + (38*5.2) + (33*5.5) + (31*5.4) + (36*5.3) + (30*5.7) = 1363.2
Σx = 35 + 32 + 38 + 33 + 31 + 36 + 30 = 235
Σy = 5.1 + 5.3 + 5.2 + 5.5 + 5.4 + 5.3 + 5.7 = 37.5
Σx^2 = (35^2) + (32^2) + (38^2) + (33^2) + (31^2) + (36^2) + (30^2) = 9615
m = (1363.2 - (235 * 37.5) / (7 * 9615) - 235^2
m ≈ -0.125
Next, let's find the y-intercept:
Using the formula for the y-intercept:
b = (Σy - m(Σx)) / n
b = (37.5 - (-0.125 * 235)) / 7
b ≈ 5.25
Therefore, the equation of the line of best fit is:
y = -0.125x + 5.25
To find how many seconds a player should expect to take to run 40 yards if they can do 22 leg-press repetitions, we substitute x = 22 into the equation:
y = -0.125(22) + 5.25
y ≈ 2.625
So the player should expect to take approximately 2.6 seconds to run 40 yards.