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 weight of a basketball player with a height of 84 inches, we can use the line of best fit for the given data.
First, let's calculate the slope and y-intercept of the line of best fit.
Using the given height and weight data, we can calculate the mean height (x̄) and mean weight (ȳ) as follows:
x̄ = (67 + 69 + 70 + 72 + 74 + 74 + 78 + 79) / 8 = 72.375 inches
ȳ = (183 + 201 + 206 + 220 + 226 + 240 + 253 + 255) / 8 = 224.125 pounds
Next, we need to calculate the sum of the products of the difference between each height (x) and the mean height (x - x̄), and the difference between each weight (y) and the mean weight (y - ȳ), denoted as ∑(x - x̄)(y - ȳ), and the sum of the squares of the difference between each height (x) and the mean height (x - x̄), denoted as ∑(x - x̄)^2:
∑(x - x̄)(y - ȳ) = (67-72.375)(183-224.125) + (69-72.375)(201-224.125) + (70-72.375)(206-224.125) + (72-72.375)(220-224.125) + (74-72.375)(226-224.125) + (74-72.375)(240-224.125) + (78-72.375)(253-224.125) + (79-72.375)(255-224.125)
= -5.375*(-41.125) + (-3.375)*(-23.125) + (-2.375)*(-18.125) + (-0.375)*(-4.125) + 1.625*(1.875) + 1.625*(15.875) + 5.625*(28.875) + 6.625*(30.875)
= 243.671875
∑(x - x̄)^2 = (67-72.375)^2 + (69-72.375)^2 + (70-72.375)^2 + (72-72.375)^2 + (74-72.375)^2 + (74-72.375)^2 + (78-72.375)^2 + (79-72.375)^2
= (-5.375)^2 + (-3.375)^2 + (-2.375)^2 + (-0.375)^2 + 1.625^2 + 1.625^2 + 5.625^2 + 6.625^2
= 66.84375
Using these values, we can calculate the slope (m) of the line of best fit:
m = ∑(x - x̄)(y - ȳ) / ∑(x - x̄)^2
= 243.671875 / 66.84375
≈ 3.63887
Next, we need to calculate the y-intercept (b) of the line of best fit:
b = ȳ - m * x̄
= 224.125 - 3.63887 * 72.375
≈ -3.69531
Therefore, the equation of the line of best fit is:
y ≈ 3.63887x - 3.69531
To predict the weight of a basketball player with a height of 84 inches, we substitute x = 84 into the equation of the line of best fit:
y ≈ 3.63887 * 84 - 3.69531
≈ 304.46228
Therefore, we would expect a basketball player with a height of 84 inches to weigh approximately 304 pounds.