To find the equation of the line of best fit, we'll use linear regression to approximate the relationship between height (independent variable) and points per game (dependent variable). Linear regression will help us find the equation of a line that best represents the given data.
Predicting a player's height if he averages 18 points per game:
Let's find the equation of the line of best fit using the given data:
height (in) 84 80 79 82 74 72 86 76
points per game 22 21 21 26 12 8 28 15
Step 1: Calculate the mean (average) of the height and points per game:
Mean of height = (84 + 80 + 79 + 82 + 74 + 72 + 86 + 76) / 8 = 79.875
Mean of points per game = (22 + 21 + 21 + 26 + 12 + 8 + 28 + 15) / 8 = 18.75
Step 2: Calculate the deviations from the mean for both height and points per game:
Deviation from mean height = height - Mean of height
Deviation from mean points per game = points per game - Mean of points per game
height (in) Deviation | points per game Deviation
84 +4.125 | 22 +3.25
80 +0.125 | 21 +2.25
79 -0.875 | 21 +2.25
82 +2.125 | 26 +7.25
74 -5.875 | 12 -6.75
72 -7.875 | 8 -10.75
86 +6.125 | 28 +9.25
76 -3.875 | 15 -3.75
Step 3: Calculate the product of the deviations for each data point:
Sum of (Deviation from mean height * Deviation from mean points per game) = (4.125 * 3.25) + (0.125 * 2.25) + (-0.875 * 2.25) + (2.125 * 7.25) + (-5.875 * -6.75) + (-7.875 * -10.75) + (6.125 * 9.25) + (-3.875 * -3.75) = 65.625
Step 4: Calculate the sum of the squared deviations for both height and points per game:
Sum of (Deviation from mean height)^2 = (4.125)^2 + (0.125)^2 + (-0.875)^2 + (2.125)^2 + (-5.875)^2 + (-7.875)^2 + (6.125)^2 + (-3.875)^2 = 183.125
Sum of (Deviation from mean points per game)^2 = (3.25)^2 + (2.25)^2 + (2.25)^2 + (7.25)^2 + (-6.75)^2 + (-10.75)^2 + (9.25)^2 + (-3.75)^2 = 273.25
Step 5: Calculate the slope (m) of the line of best fit:
m = Sum of (Deviation from mean height * Deviation from mean points per game) / Sum of (Deviation from mean height)^2
m = 65.625 / 183.125 ≈ 0.3587
Step 6: Calculate the y-intercept (b) of the line of best fit:
b = Mean of points per game - (m * Mean of height)
b = 18.75 - (0.3587 * 79.875) ≈ -5.8935
So, the equation of the line of best fit is approximately:
points per game ≈ 0.3587 * height - 5.8935
Now, let's use this equation to predict how many points per game a player averages if he is 70 inches tall.
Predicting how many points per game a player averages if he is 70 inches tall:
Using the equation of the line of best fit:
points per game ≈ 0.3587 * 70 - 5.8935
points per game ≈ 25.309
Rounded to the nearest whole number, a player who is 70 inches tall is predicted to average 25 points per game.