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?
a. 276.1 lb
b. 289.3 lb***
c. 298.4 lb
d. 307.7 lb
To estimate the weight of a basketball player with a height of 84 inches, we can create a linear regression model using the given data points. A linear regression model is used to find the relationship between two variables, in this case, height and weight.
First, we calculate the equation of the line of best fit using the given data points.
Step 1: Calculate the mean (average) of both the height and weight data points.
Mean height (h): (67 + 69 + 70 + 72 + 74 + 74 + 78 + 79) / 8 = 72.625 inches
Mean weight (w): (183 + 201 + 206 + 220 + 226 + 240 + 253 + 255) / 8 = 224.125 pounds
Step 2: Calculate the differences between each height (h) and weight (w) data points and their respective means.
differences height (dH): 67 - 72.625 = -5.625,
69 - 72.625 = -3.625,
70 - 72.625 = -2.625,
72 - 72.625 = -0.625,
74 - 72.625 = 1.375,
74 - 72.625 = 1.375,
78 - 72.625 = 5.375,
79 - 72.625 = 6.375
differences weight (dW): 183 - 224.125 = -41.125,
201 - 224.125 = -23.125,
206 - 224.125 = -18.125,
220 - 224.125 = -4.125,
226 - 224.125 = 1.875,
240 - 224.125 = 15.875,
253 - 224.125 = 28.875,
255 - 224.125 = 30.875
Step 3: Calculate the product of the height and weight differences for each data point.
product of differences (dH * dW):
(-5.625) * (-41.125) = 231.703125,
(-3.625) * (-23.125) = 83.828125,
(-2.625) * (-18.125) = 47.578125,
(-0.625) * (-4.125) = 2.578125,
(1.375) * (1.875) = 2.578125,
(1.375) * (15.875) = 21.859375,
(5.375) * (28.875) = 155.109375,
(6.375) * (30.875) = 196.828125
Step 4: Calculate the square of the height differences for each data point.
squared height differences (dH^2): (-5.625)^2 = 31.640625,
(-3.625)^2 = 13.140625,
(-2.625)^2 = 6.890625,
(-0.625)^2 = 0.390625,
(1.375)^2 = 1.890625,
(1.375)^2 = 1.890625,
(5.375)^2 = 28.890625,
(6.375)^2 = 40.640625
Step 5: Calculate the slope (b) using the formula:
b = (sum of (dH * dW)) / (sum of (dH^2))
b = (231.703125 + 83.828125 + 47.578125 + 2.578125 + 2.578125 + 21.859375 + 155.109375 + 196.828125) / (31.640625 + 13.140625 + 6.890625 + 0.390625 + 1.890625 + 1.890625 + 28.890625 + 40.640625)
b = 741.265625 / 124.372375
Step 6: Calculate the y-intercept (a) using the formula:
a = mean(weight) - (b * mean(height))
a = 224.125 - (b * 72.625)
Now, once you have the values of a and b, you can estimate the weight for a given height value by substituting the height (84 inches) into the equation:
Estimated weight = a + (b * height)
Estimated weight = [value of a] + ([value of b] * 84)
Solve the equation to get the estimated weight.
Once you have the estimated weight, you can choose the option that is closest to the calculated value.