The table below shows the height (in inches) and weight (in pounds) of eight basketball players.
Height 67
69
70 72 74 74
78
79
Weight 183 201 206 220 226 240 253 255
About how many pounds would you expect a basketball player to weigh if his height is 84 inches? Find a line of best fit for this data and use it to make your prediction.
To find a line of best fit for this data, we can use linear regression.
First, we calculate the slope and y-intercept of the line of best fit using the given height and weight data:
Sum of heights (x): 67 + 69 + 70 + 72 + 74 + 74 + 78 + 79 = 563
Sum of weights (y): 183 + 201 + 206 + 220 + 226 + 240 + 253 + 255 = 1784
Mean of heights (x̄): 563 / 8 = 70.375
Mean of weights (ȳ): 1784 / 8 = 223
Sum of (height - mean of heights)(weight - mean of weights):
(67 - 70.375)*(183 - 223) + (69 - 70.375)*(201 - 223) + (70 - 70.375)*(206 - 223) + (72 - 70.375)*(220 - 223)
+ (74 - 70.375)*(226 - 223) + (74 - 70.375)*(240 - 223) + (78 - 70.375)*(253 - 223) + (79 - 70.375)*(255 - 223) = -264.625
Sum of (height - mean of heights)^2:
(67 - 70.375)^2 + (69 - 70.375)^2 + (70 - 70.375)^2 + (72 - 70.375)^2 + (74 - 70.375)^2 + (74 - 70.375)^2
+ (78 - 70.375)^2 + (79 - 70.375)^2 = 198.75
Sum of (weight - mean of weights)^2:
(183 - 223)^2 + (201 - 223)^2 + (206 - 223)^2 + (220 - 223)^2 + (226 - 223)^2 + (240 - 223)^2 + (253 - 223)^2 + (255 - 223)^2 = 8990
Calculating the slope:
slope = (-264.625) / 198.75 ≈ -1.332
Calculating the y-intercept:
y-intercept = ȳ - slope * x̄ = 223 - (-1.332) * 70.375 ≈ 323.722
Therefore, the equation for the line of best fit is:
weight = -1.332 * height + 323.722
Now, we can use this equation to predict the weight of a basketball player with a height of 84 inches:
weight = -1.332 * 84 + 323.722 ≈ 209.712
So, we would expect a basketball player with a height of 84 inches to weigh approximately 209.712 pounds.