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 the line of best fit for this data, we can use linear regression. Here are the steps:
1. Calculate the mean (average) of the height and weight.
- Mean of height = (67 + 69 + 70 + 72 + 74 + 74 + 78 + 79) / 8 = 73.5 inches.
- Mean of weight = (183 + 201 + 206 + 220 + 226 + 240 + 253 + 255) / 8 = 223.5 pounds.
2. Calculate the difference between each height and the mean height (deviation from the mean).
- Deviations from the mean of height: -6.5, -4.5, -3.5, -1.5, 0.5, 0.5, 4.5, 5.5.
3. Calculate the difference between each weight and the mean weight (deviation from the mean).
- Deviations from the mean of weight: -40.5, -22.5, -17.5, -3.5, 2.5, 16.5, 29.5, 31.5.
4. Calculate the product of each pair of deviations (height and weight deviations).
- Product of deviations: -264.75, 101.25, 61.25, 5.25, 1.25, 8.25, 132.75, 173.25.
5. Calculate the sum of the products of deviations.
- Sum of products of deviations = -264.75 + 101.25 + 61.25 + 5.25 + 1.25 + 8.25 + 132.75 + 173.25 = 238.5.
6. Calculate the sum of squared deviations for height.
- Sum of squared deviations for height = (-6.5)^2 + (-4.5)^2 + (-3.5)^2 + (-1.5)^2 + (0.5)^2 + (0.5)^2 + (4.5)^2 + (5.5)^2 = 123.
7. Calculate the sum of squared deviations for weight.
- Sum of squared deviations for weight = (-40.5)^2 + (-22.5)^2 + (-17.5)^2 + (-3.5)^2 + (2.5)^2 + (16.5)^2 + (29.5)^2 + (31.5)^2 = 4420.
8. Calculate the slope (b) of the line of best fit.
- b = sum of products of deviations / sum of squared deviations for height
- b = 238.5 / 123 = 1.938.
9. Calculate the y-intercept (a) of the line of best fit.
- a = mean of weight - (slope * mean of height)
- a = 223.5 - (1.938 * 73.5) = 79.786.
The equation for the line of best fit is:
weight = 1.938 * height + 79.786
Now, to predict the weight when height is 84 inches, we substitute the height value into the equation:
weight = 1.938 * 84 + 79.786
weight = 162.792 + 79.786
weight ≈ 242.58 pounds.
Therefore, we can expect a basketball player with a height of 84 inches to weigh approximately 242.58 pounds.