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.
(1 point)
Responses
276.1 lb
289.3 lb
298.4 lb
307.7 lb
To find the line of best fit and make a prediction, we can use linear regression.
First, we need to calculate the slope (m) and y-intercept (b) of the line:
m = ∑((x - mean(x))(y - mean(y))) / ∑((x - mean(x))^2)
b = mean(y) - m * mean(x)
Where:
x = height
y = weight
Using the given data:
x = [67, 69, 70, 72, 74, 74, 78, 79]
y = [183, 201, 206, 220, 226, 240, 253, 255]
mean(x) = (67 + 69 + 70 + 72 + 74 + 74 + 78 + 79) / 8 = 72
mean(y) = (183 + 201 + 206 + 220 + 226 + 240 + 253 + 255) / 8 = 224.375
Σ((x - mean(x))(y - mean(y))) = [(67 - 72)(183 - 224.375)] + [(69 - 72)(201 - 224.375)] + [(70 - 72)(206 - 224.375)] + [(72 - 72)(220 - 224.375)] + [(74 - 72)(226 - 224.375)] + [(74 - 72)(240 - 224.375)] + [(78 - 72)(253 - 224.375)] + [(79 - 72)(255 - 224.375)] = -5280
Σ((x - mean(x))^2) = [(67 - 72)^2] + [(69 - 72)^2] + [(70 - 72)^2] + [(72 - 72)^2] + [(74 - 72)^2] + [(74 - 72)^2] + [(78 - 72)^2] + [(79 - 72)^2] = 136
Using these values in the formulas:
m = -5280 / 136 ≈ -38.824
b = 224.375 - (-38.824 * 72) ≈ 2598.25
So the equation of the line is y = -38.824x + 2598.25.
Now, we can substitute x = 84 into the equation to make the prediction:
y = -38.824(84) + 2598.25 ≈ 307.7 lb
Therefore, you would expect a basketball player with a height of 84 inches to weigh approximately 307.7 pounds. Therefore, the answer is 307.7 lb.