No, we cannot be confident that the predicted value will be reasonably close to the actual value.
The correlation coefficient, -0.833, indicates a strong negative correlation between the variables in the data set. However, the closeness of the predicted value to the actual value depends on the spread and variability of the data points around the regression line.
If the data points are widely spread and have a high degree of variability, even with a strong correlation, the predicted value may not be close to the actual value. In this case, the correlation coefficient alone does not provide enough information to determine the accuracy of the predicted value.
You use a line of best fit for a set of data to make a prediction about an unknown value. The correlation coefficient for your data set is Negative 0 point 833. Can you be confident that your predicted value will be reasonably close to the actual value? Why or why not?
(2 points)
3 answers
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.
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
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
To find the equation of the line of best fit, we will use the given data points to calculate the slope and y-intercept.
First, let's find the mean of the leg press repetitions and the mean of the 40-yard dash times:
Mean of leg press repetitions:
(12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 128 / 7 = 18.3
Mean of 40-yard dash times:
(8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 73.4 / 7 = 10.5
Next, let's calculate the deviation of each data point from the mean:
Deviation of leg press repetition = each value - mean of leg press repetitions:
12 - 18.3 = -6.3
32 - 18.3 = 13.7
7 - 18.3 = -11.3
11 - 18.3 = -7.3
23 - 18.3 = 4.7
28 - 18.3 = 9.7
15 - 18.3 = -3.3
Deviation of 40-yard dash time = each value - mean of 40-yard dash times:
8.6 - 10.5 = -1.9
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 = -1.0
Next, we calculate the sum of the products of the deviations of each data point:
Sum of (deviation of leg press repetition * deviation of 40-yard dash time):
(-6.3 * -1.9) + (13.7 * 4.1) + (-11.3 * -3.4) + (-7.3 * -2.2) + (4.7 * 1.4) + (9.7 * 2.9) + (-3.3 * -1.0) = 210.48
Now, we calculate the sum of the squared deviations of each variable:
Sum of (deviation of leg press repetition)^2:
(-6.3)^2 + (13.7)^2 + (-11.3)^2 + (-7.3)^2 + (4.7)^2 + (9.7)^2 + (-3.3)^2 = 589.22
Sum of (deviation of 40-yard dash time)^2:
(-1.9)^2 + (4.1)^2 + (-3.4)^2 + (-2.2)^2 + (1.4)^2 + (2.9)^2 + (-1.0)^2 = 34.74
Next, we calculate the slope of the line of best fit:
slope = (sum of (deviation of leg press repetition * deviation of 40-yard dash time)) / (sum of (deviation of leg press repetition)^2):
slope = 210.48 / 589.22 = 0.357
Finally, we calculate the y-intercept of the line of best fit:
y-intercept = mean of 40-yard dash times - (slope * mean of leg press repetitions):
y-intercept = 10.5 - (0.357 * 18.3) = 10.5 - 6.54 = 3.96
Therefore, the equation of the line of best fit is:
40-yard dash time = 0.357 * leg press repetition + 3.96
To estimate the 40-yard dash time for a player who can do 22 leg press repetitions, we substitute 22 into the equation:
40-yard dash time = 0.357 * 22 + 3.96 = 11.65
To the nearest tenth, the player should be expected to take 11.7 seconds to run 40 yards.
First, let's find the mean of the leg press repetitions and the mean of the 40-yard dash times:
Mean of leg press repetitions:
(12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 128 / 7 = 18.3
Mean of 40-yard dash times:
(8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 73.4 / 7 = 10.5
Next, let's calculate the deviation of each data point from the mean:
Deviation of leg press repetition = each value - mean of leg press repetitions:
12 - 18.3 = -6.3
32 - 18.3 = 13.7
7 - 18.3 = -11.3
11 - 18.3 = -7.3
23 - 18.3 = 4.7
28 - 18.3 = 9.7
15 - 18.3 = -3.3
Deviation of 40-yard dash time = each value - mean of 40-yard dash times:
8.6 - 10.5 = -1.9
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 = -1.0
Next, we calculate the sum of the products of the deviations of each data point:
Sum of (deviation of leg press repetition * deviation of 40-yard dash time):
(-6.3 * -1.9) + (13.7 * 4.1) + (-11.3 * -3.4) + (-7.3 * -2.2) + (4.7 * 1.4) + (9.7 * 2.9) + (-3.3 * -1.0) = 210.48
Now, we calculate the sum of the squared deviations of each variable:
Sum of (deviation of leg press repetition)^2:
(-6.3)^2 + (13.7)^2 + (-11.3)^2 + (-7.3)^2 + (4.7)^2 + (9.7)^2 + (-3.3)^2 = 589.22
Sum of (deviation of 40-yard dash time)^2:
(-1.9)^2 + (4.1)^2 + (-3.4)^2 + (-2.2)^2 + (1.4)^2 + (2.9)^2 + (-1.0)^2 = 34.74
Next, we calculate the slope of the line of best fit:
slope = (sum of (deviation of leg press repetition * deviation of 40-yard dash time)) / (sum of (deviation of leg press repetition)^2):
slope = 210.48 / 589.22 = 0.357
Finally, we calculate the y-intercept of the line of best fit:
y-intercept = mean of 40-yard dash times - (slope * mean of leg press repetitions):
y-intercept = 10.5 - (0.357 * 18.3) = 10.5 - 6.54 = 3.96
Therefore, the equation of the line of best fit is:
40-yard dash time = 0.357 * leg press repetition + 3.96
To estimate the 40-yard dash time for a player who can do 22 leg press repetitions, we substitute 22 into the equation:
40-yard dash time = 0.357 * 22 + 3.96 = 11.65
To the nearest tenth, the player should be expected to take 11.7 seconds to run 40 yards.