The table below shows the height (in inches) and weight (in pounds) of eight basketball players.
Height
67
183
69
70
72
220
74
74
78
79
Weight
201
206
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)
A-276.1 lb
B-289.3 lb
C-298.4 lb
D-307.7 lb
To find a line of best fit for this data, we can use linear regression.
First, we need to calculate the mean of the height and weight data:
Mean height: (67 + 183 + 69 + 70 + 72 + 220 + 74 + 74 + 78 + 79)/10 = 936/10 = 93.6 inches
Mean weight: (201 + 206 + 226 + 240 + 253 + 255)/6 = 1381/6 = 230.2 pounds
Next, we can calculate the slope of the line of best fit:
Slope = covariance of height and weight / variance of height
Covariance of height and weight = Σ((height - mean height) * (weight - mean weight)) / (n-1)
= (67-93.6)(201-230.2) + (183-93.6)(206-230.2) + (69-93.6)(226-230.2) + (70-93.6)(240-230.2) + (72-93.6)(253-230.2) + (220-93.6)(255-230.2) + (74-93.6)(np.nan-255) + (74-93.6)(np.nan-230.2) + (78-93.6)(np.nan-230.2) + (79-93.6)(np.nan-230.2) / (10-1)
= -4932 + 20817 + 661.6 + 619.2 + 5827.6 + 65099.2 + 1566.4 + 1410.8 + 291.2 + 726.8 / 9
= 79791.2 / 9
= 8865.7
Variance of height = Σ((height - mean height)^2) / (n-1)
= (67-93.6)^2 + (183-93.6)^2 + (69-93.6)^2 + (70-93.6)^2 + (72-93.6)^2 + (220-93.6)^2 + (74-93.6)^2 + (74-93.6)^2 + (78-93.6)^2 + (79-93.6)^2 / (10-1)
= 26214.4 + 6945.6 + 6945.6 + 6945.6 + 5807.2 + 103118.4 + 380.8 + 380.8 + 220.8 + 220.8 / 9
= 195107.2 / 9
= 21678.6
Slope = 8865.7 / 21678.6
= 0.4084
Now, we can use the slope and the mean data to find the y-intercept:
Mean weight = slope * mean height + y-intercept
230.2 = 0.4084 * 93.6 + y-intercept
230.2 = 38.21224 + y-intercept
y-intercept = 230.2 - 38.21224
y-intercept = 191.98776
Therefore, the equation of the line of best fit is:
Weight = 0.4084 * Height + 191.98776
To find the predicted weight for a height of 84 inches, we substitute this value into the equation:
Weight = 0.4084 * 84 + 191.98776
Weight = 34.3272 + 191.98776
Weight = 226.31496
Rounding to the nearest pound, we would expect a basketball player with a height of 84 inches to weigh approximately 226 pounds.
Answer: C-298.4 lb