A 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 |5.9
Am I supposed to use y=mx+b? Or????
make a scatter plot of leg press reps vs 40 yd dash time. line of best fit would be what goes through or close to most points. You have the table so try using either a graphing calculator or a program for finding the line of best fit. Try Desmos or StatCrunch
Yes, you are correct that we can use the equation of a line, y = mx + b, to find the equation of the line of best fit in this scenario. In this equation, y represents the dependent variable (40-yard dash time in seconds), x represents the independent variable (number of leg-press repetitions), m represents the slope of the line, and b represents the y-intercept.
To find the equation of the line of best fit, we need to calculate the values of m and b. Here are the steps to do it:
Step 1: Calculate the mean (average) of both the leg press repetitions and the 40-yard dash times.
For leg press repetitions:
Mean(x) = (15 + 18 + 8 + 30 + 26 + 12 + 21) / 7
Mean(x) = 130 / 7
Mean(x) ≈ 18.57
For 40-yard dash times:
Mean(y) = (5.2 + 6.3 + 6.8 + 8.2 + 8.0 + 5.3 + 5.9) / 7
Mean(y) = 45.7 / 7
Mean(y) ≈ 6.53
Step 2: Calculate the slope (m) of the line using the formula:
m = Σ((x - Mean(x)) * (y - Mean(y))) / Σ((x - Mean(x))^2)
To calculate the numerator:
Numerator = (15 - 18.57) * (5.2 - 6.53) + (18 - 18.57) * (6.3 - 6.53) + (8 - 18.57) * (6.8 - 6.53) + (30 - 18.57) * (8.2 - 6.53) + (26 - 18.57) * (8.0 - 6.53) + (12 - 18.57) * (5.3 - 6.53) + (21 - 18.57) * (5.9 - 6.53)
Numerator = (-3.57) * (-1.33) + (-0.57) * (-0.23) + (-10.57) * (0.27) + (11.43) * (1.67) + (7.43) * (1.47) + (-6.57) * (-1.23) + (2.43) * (-0.63)
Numerator = 4.7531
To calculate the denominator:
Denominator = (15 - 18.57)^2 + (18 - 18.57)^2 + (8 - 18.57)^2 + (30 - 18.57)^2 + (26 - 18.57)^2 + (12 - 18.57)^2 + (21 - 18.57)^2
Denominator = (-3.57)^2 + (-0.57)^2 + (-10.57)^2 + (11.43)^2 + (7.43)^2 + (-6.57)^2 + (2.43)^2
Denominator ≈ 790.05
Now we can calculate the slope (m):
m = Numerator / Denominator
m ≈ 4.7531 / 790.05
m ≈ 0.006
m ≈ 0.0060 (rounded to four decimal places)
Step 3: Calculate the y-intercept (b) of the line using the formula:
b = Mean(y) - (m * Mean(x))
b = 6.53 - (0.0060 * 18.57)
b = 6.53 - 0.1114
b ≈ 6.42 (rounded to two decimal places)
Therefore, the equation of the line of best fit is:
y ≈ 0.0060x + 6.42
Now, to answer your second question: If a player can do 22 leg-press repetitions, we can substitute x = 22 into the equation to find the expected 40-yard dash time (y):
y ≈ 0.0060(22) + 6.42
y ≈ 0.1320 + 6.42
y ≈ 6.552
Thus, if a player can do 22 leg-press repetitions, they can be expected to take approximately 6.552 seconds to run 40 yards.