To find the equation of the line of best fit, we need to find the slope and y-intercept.
Let's start by finding the slope. The slope is given by the formula:
slope = (Σ(x*y) - Σx*Σy/n) / (Σ(x^2) - (Σx)^2/n)
where Σ denotes the sum, x is the independent variable (number of leg-press repetitions), y is the dependent variable (40-yard dash time), and n is the number of data points.
In this case, the sum of x*y is (20*3.5) + (24*4.1) + (28*4.4) + (30*4.9) + (32*5.2) + (36*5.7) + (38*5.9) = 491.4.
The sum of x is 20 + 24 + 28 + 30 + 32 + 36 + 38 = 208.
The sum of y is 3.5 + 4.1 + 4.4 + 4.9 + 5.2 + 5.7 + 5.9 = 34.7.
The sum of x^2 is 20^2 + 24^2 + 28^2 + 30^2 + 32^2 + 36^2 + 38^2 = 7240.
Now we can substitute these values into the slope formula:
slope = (491.4 - (208*34.7)/7) / (7240 - (208^2)/7)
= (491.4 - 6065.6/7) / (7240 - 17312/7)
= (491.4 - 866.514) / (7240 - 2473.143)
= -375.114 / 4766.857
= -0.0788 (rounded to four decimal places)
Now let's find the y-intercept using the formula:
y-intercept = mean(y) - slope * mean(x)
The mean of x is (20 + 24 + 28 + 30 + 32 + 36 + 38) / 7 = 29.714.
The mean of y is (3.5 + 4.1 + 4.4 + 4.9 + 5.2 + 5.7 + 5.9) / 7 = 4.957.
Substituting these values into the y-intercept formula:
y-intercept = 4.957 - (-0.0788 * 29.714)
= 4.957 + 2.3369
= 7.2939 (rounded to four decimal places)
Therefore, the equation of the line of best fit is:
y = -0.0788x + 7.2939
To find out how many seconds a player should expect to take to run 40 yards if they can do 22 leg-press repetitions, we can substitute x = 22 into the equation:
y = -0.0788 * 22 + 7.2939
= -1.7316 + 7.2939
= 5.5623 (rounded to four decimal places)
Therefore, a player who can do 22 leg-press repetitions should expect to take approximately 5.6 seconds to run 40 yards.
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.
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