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.
X = 12, 32, 7, 11, 23, 28, 15
Y = 8.6, 14.6, 7.1, 8.3, 11.9, 13.4, 9.5
To find the equation of the line of best fit, we need to find the slope and the y-intercept.
First, let's find the slope (m) using the formula:
m = (ΣXY - ΣXΣY/n) / (ΣX^2 - (ΣX)^2/n)
Where:
ΣXY = sum of the product of each X value and corresponding Y value (X1Y1 + X2Y2 + ...)
ΣX = sum of all X values (X1 + X2 + ...)
ΣY = sum of all Y values (Y1 + Y2 + ...)
ΣX^2 = sum of the squares of all X values (X1^2 + X2^2 + ...)
n = number of data points
ΣXY = (12 * 8.6) + (32 * 14.6) + (7 * 7.1) + (11 * 8.3) + (23 * 11.9) + (28 * 13.4) + (15 * 9.5) = 967.2
ΣX = 12 + 32 + 7 + 11 + 23 + 28 + 15 = 128
ΣY = 8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5 = 73.4
ΣX^2 = (12^2) + (32^2) + (7^2) + (11^2) + (23^2) + (28^2) + (15^2) = 2981
n = 7
Substituting the values into the formula:
m = (967.2 - (128 * 73.4)/7) / (2981 - (128^2)/7)
m = (967.2 - 128 * 73.4/7) / (2981 - 128^2/7)
m = (967.2 - 128(73.4)/7) / (2981 - 128^2/7)
m = (967.2 - 128(73.4)/7) / (2981 - 128^2/7)
m = (967.2 - 128(73.4)/7) / (2981 - 128^2/7)
m = (967.2 - 942.4) / (2981 - 2048/7)
m = 24.8 / 33.4
m = 0.74251497 (rounded to 8 decimal places)
Next, let's find the y-intercept (b) using the formula:
b = (ΣY - mΣX) / n
Substituting the values into the formula:
b = (73.4 - 0.74251497 * 128) / 7
b = (73.4 - 94.99983936) / 7
b = -21.59983936/7
b = -3.08569134 (rounded to 8 decimal places)
Now we have the equation of the line of best fit:
y = mx + b
Substituting the values we found:
y = 0.74251497x - 3.08569134
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 = 0.74251497(22) - 3.08569134
y = 16.3373286 - 3.08569134
y ≈ 13.25163726
Therefore, the player should expect to take approximately 13.3 seconds to run 40 yards.