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.
Leg Press (reps)
12
32
7
11
23
28
15
40-yard Dash (s)
8.6
14.6
7.1
8.3
11.9
13.4
9.5
(2 points)
To find the equation of the line of best fit, we need to calculate the slope and y-intercept using the given data points.
First, let's calculate the average number of leg-press repetitions and the average 40-yard dash time:
Average leg press repetitions: (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 18.6
Average 40-yard dash time: (8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.5
Next, we need to calculate the differences from the averages for each data point:
Leg press difference: 12 - 18.6 = -6.6
40-yard dash time difference: 8.6 - 10.5 = -1.9
Repeat this calculation for all data points:
- Leg press difference: 32 - 18.6 = 13.4, 7 - 18.6 = -11.6, 11 - 18.6 = -7.6, 23 - 18.6 = 4.4, 28 - 18.6 = 9.4, 15 - 18.6 = -3.6.
- 40-yard dash time difference: 14.6 - 10.5 = 4.1, 7.1 - 10.5 = -3.4, 8.3 - 10.5 = -2.2, 11.9 - 10.5 = 1.4, 13.4 - 10.5 = 2.9, 9.5 - 10.5 = -1.
Next, we need to square the leg press differences:
- Squared leg press difference: (-6.6)^2 = 43.6, (-11.6)^2 = 134.6, (-7.6)^2 = 57.76, (4.4)^2 = 19.36, (9.4)^2 = 88.36, (-3.6)^2 = 12.96.
Now, we multiply the squared leg press differences by the 40-yard dash time differences:
- Product: 43.6 * -1.9 = -82.84, 134.6 * -3.4 = -457.24, 57.76 * -2.2 = -127.072, 19.36 * 1.4 = 27.104, 88.36 * 2.9 = 255.644, 12.96 * -1 = -12.96.
Finally, we need to find the sum of the products and the sum of the squared leg press differences:
- Sum of the products: -82.84 + -457.24 + -127.072 + 27.104 + 255.644 + -12.96 = -397.388.
- Sum of the squared leg press differences: 43.6 + 134.6 + 57.76 + 19.36 + 88.36 + 12.96 = 356.64.
Now we can calculate the slope of the line:
Slope = Sum of the products / Sum of the squared leg press differences
= -397.388 / 356.64
= -1.12
Next, we can calculate the y-intercept:
y-intercept = Average 40-yard dash time - Slope * Average leg press repetitions
= 10.5 - (-1.12 * 18.6)
= 10.5 + 20.7936
= 31.3 (rounded to the nearest tenth)
Therefore, the equation of the line of best fit is:
40-yard dash time = -1.12 * leg press repetitions + 31.3
To predict how many seconds a player will take to run 40 yards if they can do 22 leg-press repetitions:
40-yard dash time = -1.12 * 22 + 31.3
= -24.64 + 31.3
= 6.66 (rounded to the nearest tenth)
Therefore, the player should be expected to take approximately 6.7 seconds to run 40 yards.