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 need to calculate the slope and y-intercept.
Step 1: Calculate the average number of leg-press repetitions and the average 40-yard dash time for the seven players.
Player 1: 18 leg-press repetitions, 4.8 seconds
Player 2: 19 leg-press repetitions, 4.9 seconds
Player 3: 20 leg-press repetitions, 5.1 seconds
Player 4: 22 leg-press repetitions, 5.3 seconds
Player 5: 21 leg-press repetitions, 5.2 seconds
Player 6: 19 leg-press repetitions, 5 seconds
Player 7: 20 leg-press repetitions, 5.1 seconds
Average leg-press repetitions: (18 + 19 + 20 + 22 + 21 + 19 + 20) / 7 = 19.857
Average 40-yard dash time: (4.8 + 4.9 + 5.1 + 5.3 + 5.2 + 5 + 5.1) / 7 = 5
Step 2: Calculate the differences from the average values for each player.
Player 1: -1.857 leg-press repetitions, -0.2 seconds
Player 2: -0.857 leg-press repetitions, -0.1 seconds
Player 3: 0.143 leg-press repetitions, 0.1 seconds
Player 4: 2.143 leg-press repetitions, 0.3 seconds
Player 5: 1.143 leg-press repetitions, 0.2 seconds
Player 6: -0.857 leg-press repetitions, 0 seconds
Player 7: 0.143 leg-press repetitions, 0.1 seconds
Step 3: Calculate the sum of the products of the differences from the average for leg-press repetitions and 40-yard dash times.
Sum of products = (-1.857 * -0.2) + (-0.857 * -0.1) + (0.143 * 0.1) + (2.143 * 0.3) + (1.143 * 0.2) + (-0.857 * 0) + (0.143 * 0.1)
= 0.3714
Step 4: Calculate the sum of the squares of the differences from the average for leg-press repetitions.
Sum of squares of differences = (-1.857)^2 + (-0.857)^2 + (0.143)^2 + (2.143)^2 + (1.143)^2 + (-0.857)^2 + (0.143)^2
= 12.9182
Step 5: Calculate the slope.
slope = (sum of products) / (sum of squares of differences)
= 0.3714 / 12.9182
= 0.0287
Step 6: Calculate the y-intercept.
y-intercept = average 40-yard dash time - (slope * average leg-press repetitions)
= 5 - (0.0287 * 19.857)
= 4.4270
Therefore, the equation of the line of best fit is:
y = 0.0287x + 4.4270
To find how many seconds it should take a player to run 40 yards if they can do 22 leg-press repetitions:
y = 0.0287 * 22 + 4.4270
= 5.9134
Therefore, the player should take approximately 5.9 seconds to run 40 yards.