Question

the scatter plot and line of best fit below show the length of 14 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 height for someone with a femur length of 45 cm?

Explanation:

Step1: Find the slope of the line of best fit

We can use two points on the line, say \((30, 127)\) and \((35, 139)\). The slope \(m\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{139 - 127}{35 - 30}=\frac{12}{5} = 2.4\).

Step2: Find the equation of the line

Using the point - slope form \(y - y_1=m(x - x_1)\) with the point \((30,127)\) and \(m = 2.4\).
\(y-127=2.4(x - 30)\)
\(y-127 = 2.4x-72\)
\(y=2.4x + 55\) (Wait, let's check with another point. Let's take \((35,139)\), substituting \(x = 35\) into \(y=2.4x + 55\), we get \(y=2.4\times35+55=84 + 55=139\), which matches. Another point \((40,151)\): \(y=2.4\times40+55=96 + 55 = 151\), which also matches).

Step3: Predict the height for \(x = 45\)

Substitute \(x = 45\) into the equation \(y=2.4x + 55\)
\(y=2.4\times45+55\)
\(y = 108+55\)
\(y=163\)

Wait, let's re - calculate the slope again. Let's use \((30,127)\) and \((40,151)\). The slope \(m=\frac{151 - 127}{40 - 30}=\frac{24}{10}=2.4\), which is correct. The equation of the line is \(y - 127=2.4(x - 30)\), so \(y=2.4x-72 + 127=2.4x + 55\). When \(x = 45\), \(y=2.4\times45+55=108 + 55=163\).

Alternatively, we can use the two - point formula with \((35,139)\) and \((40,151)\). The slope \(m=\frac{151 - 139}{40 - 35}=\frac{12}{5}=2.4\). The equation is \(y-139 = 2.4(x - 35)\), \(y-139=2.4x-84\), \(y=2.4x + 55\). Same as before.

So when \(x = 45\), \(y=2.4\times45+55 = 163\).

Answer:

\(163\)