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 the data and make a prediction, we can use linear regression.
Step 1: Calculate the mean height and weight.
Mean height = (67 + 69 + 70 + 72 + 74 + 74 + 78 + 79) / 8 = 72.125 inches
Mean weight = (183 + 201 + 206 + 220 + 226 + 240 + 253 + 255) / 8 = 221.5 pounds
Step 2: Calculate the deviations from the mean for both height and weight.
Deviation from mean height = Height - Mean height
Deviation from mean weight = Weight - Mean weight
Height Deviation from Mean Height Weight Deviation from Mean Weight
67 -5.125 183 -38.5
69 -3.125 201 -20.5
70 -2.125 206 -15.5
72 -0.125 220 -1.5
74 1.875 226 4.5
74 1.875 240 18.5
78 5.875 253 31.5
79 6.875 255 33.5
Step 3: Calculate the product of the deviations from the mean for both height and weight.
Product of deviations = Deviation from Mean Height * Deviation from Mean Weight
Height Deviation from Mean Height Weight Deviation from Mean Weight Product of Deviations
67 -5.125 183 -38.5 197.3125
69 -3.125 201 -20.5 64.0625
70 -2.125 206 -15.5 32.8125
72 -0.125 220 -1.5 0.1875
74 1.875 226 4.5 8.4375
74 1.875 240 18.5 34.6875
78 5.875 253 31.5 184.6875
79 6.875 255 33.5 230.9375
Step 4: Calculate the squared deviations from the mean for height.
Squared deviation for height = (Deviation from Mean Height)^2
Height Deviation from Mean Height Squared Deviation for Height
67 -5.125 26.265625
69 -3.125 9.765625
70 -2.125 4.515625
72 -0.125 0.015625
74 1.875 3.515625
74 1.875 3.515625
78 5.875 34.515625
79 6.875 47.265625
Step 5: Calculate the sum of the squared deviations for height.
Sum of squared deviations for height = Σ(Squared deviation for height)
Sum of squared deviations for height = 26.265625 + 9.765625 + 4.515625 + 0.015625 + 3.515625 + 3.515625 + 34.515625 + 47.265625
= 129.375
Step 6: Calculate the product of deviations from the mean for height and weight.
Sum of product of deviations = Σ(Product of deviations)
Sum of product of deviations = 197.3125 + 64.0625 + 32.8125 + 0.1875 + 8.4375 + 34.6875 + 184.6875 + 230.9375
= 753.125
Step 7: Calculate the slope of the line of best fit.
Slope = Sum of product of deviations / Sum of squared deviations for height
Slope = 753.125 / 129.375
= 5.825
Step 8: Calculate the y-intercept of the line of best fit.
y-intercept = Mean weight - (Slope * Mean height)
y-intercept = 221.5 - (5.825 * 72.125)
= 221.5 - 419.90625
= -198.40625
Step 9: Write the equation of the line of best fit.
y = mx + b
Weight = (Slope * Height) + y-intercept
Weight = 5.825 * Height - 198.40625
Step 10: Use the equation to make a prediction for a height of 84 inches.
Weight = 5.825 * 84 - 198.40625
Weight = 489.3 - 198.40625
Weight = 290.89375
Therefore, we would expect a basketball player with a height of 84 inches to weigh approximately 291 pounds.