plot leg presses on the x-axis , and the corresponding 40 yd dash times on the y-axis
then fit a line to the points (best fit)
A college football coach wants to know if there 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 time each player can leg press 350 lbs. 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 for seven randomly selected players. What is the equation of the line of best fit? How many second should he expect a player to take a run 40 yards if that player can do 22 leg press repetitions. Round any values to the nearest tenth if necessary.
leg press 15 18 8 30 26 12 21
40 yard dash 5.2 6.3 6.8 8.2 8.2 5.3 5.9
then fit a line to the points (best fit)
It appears to me that the results of the experiment are all over the place. The two slowest times were recorded when the athlete's leg strength was the greatest and the fastest times when the number of leg presses were the least. Looks like an inverse relation: the weaker the legs, the faster the times? This does not seem logical.
Plotting the points does not help much, it shows no nice pattern.
Good luck getting any valid statistical results from this.
stronger (and larger) linemen are usually slower than the smaller backs and receivers
Step 1: Calculate the mean of the leg press repetitions and the mean of the 40-yard dash times.
Mean of leg press repetitions:
(15 + 18 + 8 + 30 + 26 + 12 + 21) / 7 = 20.7
Mean of 40-yard dash times:
(5.2 + 6.3 + 6.8 + 8.2 + 8.2 + 5.3 + 5.9) / 7 = 6.6714 (rounded to four decimal places)
Step 2: Calculate the sum of the products of each leg press repetition and its corresponding 40-yard dash time, as well as the sum of the squares of the leg press repetitions.
Sum of the products:
(15 * 5.2) + (18 * 6.3) + (8 * 6.8) + (30 * 8.2) + (26 * 8.2) + (12 * 5.3) + (21 * 5.9) = 1936.5
Sum of the squares of the leg press repetitions:
(15^2) + (18^2) + (8^2) + (30^2) + (26^2) + (12^2) + (21^2) = 3030
Step 3: Calculate the slope (m) of the line of best fit using the formula:
m = (n * ∑(x*y) - ∑(x) * ∑(y)) / (n * ∑(x^2) - (∑(x))^2)
where n is the number of data points, x is the leg press repetitions, y is the 40-yard dash times, ∑ represents the sum, and ∑(x*y) represents the sum of the products of x and y.
n = 7
∑(x*y) = 1936.5
∑(x) = 20.7 * 7 = 144.9
∑(x^2) = 3030
Substituting the values into the formula:
m = (7 * 1936.5 - 144.9 * 6.6714) / (7 * 3030 - 144.9^2)
m = 0.053
Step 4: Calculate the y-intercept (b) of the line of best fit using the formula:
b = (∑(y) - m * ∑(x)) / n
∑(y) = 6.6714 * 7 = 46.7
Substituting the values into the formula:
b = (46.7 - 0.053 * 144.9) / 7
b = -1.920
The equation of the line of best fit is y = 0.053x - 1.920.
To calculate the expected time for a player to run 40 yards if that player can do 22 leg press repetitions, substitute x = 22 into the equation and solve for y:
y = 0.053 * 22 - 1.920
y = 0.166
Therefore, the player should be expected to take approximately 0.2 seconds (rounded to the nearest tenth) to run 40 yards.