Question

the scatter plot and line of best fit below show the length of 10 peoples femur (the long leg bone in the thigh) and their height in centimeters. based on the line of best fit, what would be the predicted femur length for someone with a height of 187 cm?
(scatter plot with x-axis labeled femur length (centimeters) from 30 to 41, y-axis labeled height (centimeters) from 127 to 160. line of best fit passes through (30,127), (35,139), (40,151))

Explanation:

Step1: Find the slope of the line of best fit

We have two points on the line of best fit: \((30, 127)\) and \((35, 139)\) and \((40, 151)\). Let's use \((30, 127)\) and \((40, 151)\) to calculate the slope \(m\). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Substituting \(x_1 = 30,y_1 = 127,x_2 = 40,y_2 = 151\), we get \(m=\frac{151 - 127}{40 - 30}=\frac{24}{10}=2.4\). Wait, no, wait, actually, looking at the axes: the \(x\)-axis is Femur Length (centimeters) and \(y\)-axis is Height (centimeters). So the points are \((x_1,y_1)=(30,127)\), \((x_2,y_2)=(35,139)\), \((x_3,y_3)=(40,151)\). Let's recalculate the slope between \((30,127)\) and \((35,139)\): \(m=\frac{139 - 127}{35 - 30}=\frac{12}{5}=2.4\). Between \((35,139)\) and \((40,151)\): \(\frac{151 - 139}{40 - 35}=\frac{12}{5}=2.4\). So the slope \(m = 2.4\) (height per femur length, or actually, the line is \(y=mx + b\), where \(x\) is femur length, \(y\) is height). Wait, no, let's find the equation of the line. Using point \((30,127)\) and slope \(m = 2.4\). The equation of a line is \(y - y_1=m(x - x_1)\). So \(y-127 = 2.4(x - 30)\). Simplifying, \(y=2.4x-72 + 127=2.4x + 55\)? Wait, no, wait, when \(x = 30\), \(2.4\times30=72\), \(72 + 55=127\), yes. Let's check with \(x = 35\): \(2.4\times35=84\), \(84 + 55=139\), correct. With \(x = 40\): \(2.4\times40 = 96\), \(96+55 = 151\), correct. So the equation of the line of best fit is \(y = 2.4x+55\), where \(x\) is femur length (in cm) and \(y\) is height (in cm). But we need to find the femur length (\(x\)) when height (\(y\)) is 187 cm. So we solve for \(x\) in the equation \(y=2.4x + 55\).

Step2: Solve for \(x\) when \(y = 187\)

Substitute \(y = 187\) into the equation \(187=2.4x + 55\). Subtract 55 from both sides: \(187 - 55=2.4x\), so \(132 = 2.4x\). Then divide both sides by 2.4: \(x=\frac{132}{2.4}=55\). Wait, let's check the calculation: \(2.4\times55 = 132\), \(132+55 = 187\), correct. Wait, but let's verify the slope again. Wait, maybe I mixed up \(x\) and \(y\). Wait, the \(x\)-axis is Femur Length (centimeters), \(y\)-axis is Height (centimeters). So the line is height as a function of femur length: \(y = mx + b\). The points are (femur length, height): (30,127), (35,139), (40,151). So the change in \(y\) (height) is \(139 - 127 = 12\) when femur length changes by \(35 - 30 = 5\), so slope \(m=\frac{12}{5}=2.4\) (height per femur length). So the equation is \(y=2.4x + b\). Using (30,127): \(127=2.4\times30 + b\), \(127 = 72 + b\), so \(b = 127 - 72 = 55\). So equation is \(y = 2.4x + 55\). Now, we need to find \(x\) (femur length) when \(y = 187\) (height). So \(187=2.4x + 55\). Subtract 55: \(132=2.4x\). Divide by 2.4: \(x=\frac{132}{2.4}\). Let's compute that: \(132\div2.4\). Multiply numerator and denominator by 10: \(1320\div24 = 55\). So the predicted femur length is 55 centimeters.

Answer:

55