A college football coach wants to know if the 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 times each player can leg press 350 pounds. The following week, he has each player sprint 40 yards every day. The tables 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?
Leg Press (reps)|40-yard Dash (s)
15 |5.2
18 |6.3
8 |6.8
30 |8.2
26 |8.0
12 |5.3
21 |5.9
Can someone tell me how to figure out the equation of the line? I really don't understand.
To figure out the equation of the line of best fit, you can use linear regression. Linear regression is a statistical technique that models the relationship between two variables by fitting a linear equation to observed data points.
First, you need to calculate the correlation coefficient, which measures the strength and direction of the relationship between the two variables. In this case, the variables are the number of leg press repetitions and the corresponding 40-yard dash time.
Next, calculate the slope and intercept of the line of best fit. The slope represents how much the 40-yard dash time changes for each additional leg press repetition, while the intercept represents the starting point of the line.
Here are the steps to calculate the equation of the line of best fit:
1. Calculate the correlation coefficient (r). If you have statistical software or a calculator, you can use the CORREL function. Otherwise, you can use a formula. The formula for r is:
r = [n∑(xy) - (∑x)(∑y)] / sqrt{ [n∑x^2− (∑x)^2] [n∑y^2− (∑y)^2] }
In this formula:
- n: the number of data points (in this case, 7)
- ∑: the sum of the values
- x: the leg press repetitions
- y: the 40-yard dash time
2. Calculate the slope (m) using the formula:
m = r * (sy / sx)
In this formula:
- r: the correlation coefficient
- sy: the standard deviation of the 40-yard dash time
- sx: the standard deviation of the leg press repetitions
3. Calculate the intercept (b) using the formula:
b = y̅ - m * x̅
In this formula:
- y̅: the mean (average) of the 40-yard dash times
- x̅: the mean (average) of the leg press repetitions
4. Once you have the slope (m) and intercept (b), you can write the equation of the line in the form:
y = mx + b
Now let's go through the calculations for the given data:
1. Calculate the sum of leg press repetitions (∑x), the sum of 40-yard dash times (∑y), the sum of the products of leg press repetitions and 40-yard dash times (∑xy), the sum of the squares of leg press repetitions (∑x^2), and the sum of the squares of 40-yard dash times (∑y^2).
2. Use the calculations from step 1 to calculate the correlation coefficient (r) using the given formula.
3. Calculate the standard deviation of the leg press repetitions (sx) and the standard deviation of the 40-yard dash times (sy) using the following formulas:
sx = sqrt{ (∑x^2 / n) - (x̅)^2 }
sy = sqrt{ (∑y^2 / n) - (y̅)^2 }
4. Calculate the slope (m) using the given formula.
5. Calculate the intercept (b) using the given formula.
Now that you have the slope (m) and intercept (b), you can write the equation of the line of best fit:
y = mx + b
In this equation, y represents the 40-yard dash time and x represents the number of leg press repetitions.
To find out how many seconds a player would be expected to take to run 40 yards if they can do 22 leg press repetitions, substitute 22 for x in the equation and solve for y.
I hope this explanation helps you understand how to figure out the equation of the line of best fit!