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.l
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To find the line of best fit for this data, you can use linear regression. Linear regression is a statistical method that finds the equation of a straight line that best fits the data points.
Let's first calculate the mean height and mean weight of the basketball players:
Mean height = (67 + 69 + 70 + 72 + 74 + 74 + 78 + 79) / 8 = 72.25 inches
Mean weight = (183 + 201 + 206 + 220 + 226 + 240 + 253 + 255) / 8 = 224.375 pounds
Next, calculate the deviations from the mean for both height and weight:
Height deviation = height - mean height
Weight deviation = weight - mean weight
Using the given data, we get:
| Height | Weight | Height deviation | Weight deviation |
|--------|--------|-----------------|-----------------|
| 67 | 183 | -5.25 | -41.375 |
| 69 | 201 | -3.25 | -23.375 |
| 70 | 206 | -2.25 | -18.375 |
| 72 | 220 | -0.25 | -4.375 |
| 74 | 226 | 1.75 | 1.625 |
| 74 | 240 | 1.75 | 15.625 |
| 78 | 253 | 5.75 | 28.625 |
| 79 | 255 | 6.75 | 30.625 |
Next, calculate the product of the height deviations and weight deviations:
Height deviation * Weight deviation = (-5.25 * -41.375) + (-3.25 * -23.375) + (-2.25 * -18.375) + (-0.25 * -4.375) + (1.75 * 1.625) + (1.75 * 15.625) + (5.75 * 28.625) + (6.75 * 30.625) = 1770.0625
Next, calculate the squared height deviations:
Height deviation^2 = (-5.25)^2 + (-3.25)^2 + (-2.25)^2 + (-0.25)^2 + (1.75)^2 + (1.75)^2 + (5.75)^2 + (6.75)^2 = 143.75
Using these values, you can find the slope of the line of best fit, which is given by:
Slope = Height deviation * Weight deviation / Height deviation^2 = 1770.0625 / 143.75 = 12.2989
Next, use the formula for the equation of a straight line:
y = mx + b
where y is the weight, x is the height, m is the slope, and b is the y-intercept.
To find the y-intercept, substitute the mean values of height and weight into the equation:
224.375 = 12.2989 * 72.25 + b
b = 224.375 - (12.2989 * 72.25)
b = 224.375 - 887.375525
b = -663
The equation of the line of best fit becomes:
y = 12.2989x - 663
Now, to predict the weight of a basketball player with a height of 84 inches, substitute x = 84 into the equation:
y = 12.2989 * 84 - 663
y = 1033.4216 - 663
y = 370.4216
Therefore, you would expect a basketball player with a height of 84 inches to weigh approximately 370.42 pounds, according to the line of best fit.