Question

the table shows the femur lengths (in centimeters) and heights (in centimeters) of several people. do the data show a linear relationship? if so, write an equation of a line of fit and use it to (a) estimate the height of a person whose femur is 35 centimeters long? (b) estimate the femur length if a persons height is 190 centimeters.

femur length (cm) x	height (cm) y
45	183
32	151
50	195
37	162
41	174
30	141
34	151
47	185
45	182

equations:

show the work for (a)

show the work for (b)

Explanation:

Step1: Calculate slope and y - intercept

Let \(x\) be femur length and \(y\) be height. First, find the mean of \(x\) values \(\bar{x}=\frac{40 + 45+32+50+37+41+30+34+47+45}{10}=39.1\) and the mean of \(y\) values \(\bar{y}=\frac{170 + 183+151+195+162+174+141+151+185+182}{10}=169.4\).
The slope \(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\).
\(\sum_{i = 1}^{10}(x_{i}-\bar{x})(y_{i}-\bar{y})=(40 - 39.1)(170 - 169.4)+(45 - 39.1)(183 - 169.4)+\cdots+(45 - 39.1)(182 - 169.4)=574.9\)
\(\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=(40 - 39.1)^{2}+(45 - 39.1)^{2}+\cdots+(45 - 39.1)^{2}=288.9\)
\(m=\frac{574.9}{288.9}\approx2\)
The y - intercept \(b=\bar{y}-m\bar{x}=169.4-2\times39.1 = 169.4 - 78.2=91.2\).
The equation of the line of fit is \(y = 2x+91.2\).

Step2: Solve part (a)

Substitute \(x = 35\) into \(y = 2x+91.2\).
\(y=2\times35 + 91.2=70+91.2 = 161.2\) cm.

Step3: Solve part (b)

Set \(y = 190\) in \(y = 2x+91.2\).
\(190=2x+91.2\)
\(2x=190 - 91.2\)
\(2x=98.8\)
\(x = 49.4\) cm.

Answer:

Equations: \(y = 2x+91.2\)
(a) 161.2 cm
(b) 49.4 cm