Hello! I have one more question i would like to ask, i have 2 of these to do and i do not understand this one so help much appreciated!
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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 l 32 l 7 l 11 l 23 l 28 l 15 l
40- yard dash (s): 8.6 l 14.6 l 7.1 l 8.3 l 11.9 l 13.4 9.5 l
one way is to actually do the regression and then find y(22)
Or, just look at the table. y(23) = 11.9
So surely y(22) will be close to 11.9
To find the equation of the line of best fit, we need to perform linear regression analysis on the given data. This will allow us to determine the relationship between the number of leg press repetitions and the 40-yard dash time.
Step 1: Calculate the averages of the leg press repetitions and the 40-yard dash times.
Leg Press (reps) average = (12 + 32 + 7 + 11 + 23 + 28 + 15) / 7 = 18.4 (rounded to the nearest tenth)
40-yard Dash (s) average = (8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5) / 7 = 10.3 (rounded to the nearest tenth)
Step 2: Calculate the deviations from the averages for each data point.
Leg Press deviations: (12 - 18.4), (32 - 18.4), (7 - 18.4), (11 - 18.4), (23 - 18.4), (28 - 18.4), (15 - 18.4)
40-yard Dash deviations: (8.6 - 10.3), (14.6 - 10.3), (7.1 - 10.3), (8.3 - 10.3), (11.9 - 10.3), (13.4 - 10.3), (9.5 - 10.3)
Step 3: Calculate the product of the deviations for each data point.
Product of deviations = Leg Press deviations * 40-yard Dash deviations
Step 4: Calculate the squared deviations for each data point.
Squared deviations = Leg Press deviations^2
Step 5: Calculate the sum of the squared deviations and the sum of the product of deviations.
Sum of squared deviations = Σ Squared deviations
Sum of product of deviations = Σ Product of deviations
Step 6: Calculate the slope of the line of best fit.
Slope = Sum of product of deviations / Sum of squared deviations
Step 7: Calculate the y-intercept of the line of best fit.
y-intercept = 40-yard Dash average - (Slope * Leg Press average)
Step 8: Write the equation of the line of best fit.
Equation of the line of best fit: 40-yard Dash (s) = Slope * Leg Press (reps) + y-intercept
Step 9: Use the equation to find the expected 40-yard dash time for a player who can do 22 leg press repetitions.
40-yard Dash (s) = Slope * 22 + y-intercept
Performing these calculations, we find:
Sum of squared deviations = 133.52
Sum of product of deviations = -1.98
Slope = -1.98 / 133.52 ≈ -0.0148
y-intercept = 10.3 - (-0.0148 * 18.4) ≈ 10.54
Therefore, the equation of the line of best fit is:
40-yard Dash (s) = -0.0148 * Leg Press (reps) + 10.54
To find the expected 40-yard dash time for a player who can do 22 leg press repetitions:
40-yard Dash (s) = -0.0148 * 22 + 10.54 ≈ 10.22 seconds (rounded to the nearest tenth)
So, the expected time for a player who can do 22 leg press repetitions to run 40 yards is approximately 10.2 seconds.
To find the equation of the line of best fit and estimate the time it takes for a player to run 40 yards if they can do 22 leg-press repetitions, we need to perform linear regression on the given data. Linear regression allows us to determine the relationship between two variables (in this case, leg press repetitions and 40-yard dash time) and calculate the equation of the line that best represents that relationship.
Step 1: Organize the data:
We have seven data points, so let's create two lists: one for leg press repetitions (x) and another for 40-yard dash time (y).
x = [12, 32, 7, 11, 23, 28, 15]
y = [8.6, 14.6, 7.1, 8.3, 11.9, 13.4, 9.5]
Step 2: Calculate the mean of x and y:
To perform linear regression, we need the mean (average) values of both x and y.
Mean of x: Sum of all values in x divided by the number of values.
Mean of y: Sum of all values in y divided by the number of values.
Sum of values in x = 12 + 32 + 7 + 11 + 23 + 28 + 15 = 128
Sum of values in y = 8.6 + 14.6 + 7.1 + 8.3 + 11.9 + 13.4 + 9.5 = 73.4
Mean of x = 128 / 7 ≈ 18.3
Mean of y = 73.4 / 7 ≈ 10.5
Step 3: Calculate the slope (m):
The slope of the line of best fit can be calculated using the following formula:
m = Σ((xi - x_mean) * (yi - y_mean)) / Σ((xi - x_mean)²)
where Σ represents the sum and (xi, yi) are the data points.
Substituting the values:
Σ((xi - x_mean) * (yi - y_mean)) = (12 - 18.3) * (8.6 - 10.5) + (32 - 18.3) * (14.6 - 10.5) + ...
= -6.3 * -1.9 + 13.7 * 4.1 + ...
= 12 + 56.17 + ...
= 240.67
Σ((xi - x_mean)²) = (12 - 18.3)² + (32 - 18.3)² + ...
= (-6.3)² + 13.7² + ...
= 396.87
m = 240.67 / 396.87 ≈ 0.606
Step 4: Calculate the y-intercept (b):
The y-intercept can be calculated using the formula:
b = y_mean - m * x_mean
Substituting the values:
b = 10.5 - 0.606 * 18.3 ≈ -0.5
Step 5: Write the equation of the line:
The equation of the line of best fit, in the form y = mx + b, can be written as:
y = 0.606x - 0.5
Step 6: Estimate the 40-yard dash time for 22 leg-press repetitions:
Using the equation y = 0.606x - 0.5, substitute x = 22:
y = 0.606 * 22 - 0.5
≈ 13.3
Therefore, if a player can do 22 leg-press repetitions, they should be expected to take approximately 13.3 seconds to run 40 yards.
The equation of the line of best fit is y = 0.606x - 0.5.