Chapter 3, Problem 3.35SE

Armspan and Height Leonardo da Vinci (1452-1519) drew a sketch of a man, indication that a person’s armspan (meausring across the back with arms outstretched to make a “T”) is roughly equal to the person’s height. To test this claim, we measured eight people with the following results:

1. Draw sactterplot for armspan and height. Use the same scale on both the horizontal and vertical axes. Describe the relationship between the two variables.
2. Calculate the correlation coefficient relating armspan and height.
3. If you were to calculate the regression line for predicting height based on a person’s armspan, how would you estimate the slope of this line?
4. Find the regression line relating armspan to a person’s height.
5. If a person has an armspan of 62 inches, what would you predict the person’s height to be?

To determine

Draw a scatterplot for the given relationship.

Answer to Problem 3.35SE

The scatterplot is.

  

Explanation of Solution

Given:

The table shows the relation betweeneight people’s armspan and height.

  

Calculation:

The scatterplot for the given points are.

  

The horizontal axis x represents armspn and the vertical axis y represents the height.

To determine

Find the correlation coefficient using the given points.

Answer to Problem 3.35SE

The correlation coefficient is $r\approx 0.95$ .

Explanation of Solution

Given:

The table shows the relation between eight people’s armspan and height.

  

Calculation:

First determine ${\displaystyle \sum x_i}{y}_i$ , ${\displaystyle \sum x_i}$ and ${\displaystyle \sum y_i}$ .

  $x_i\to$ first coordinates of the ordered pairs.

  $y_i\to$ second coordinates of the ordered pairs.

  $\begin{array}{l}{\displaystyle \sum x_i}{y}_i=68\cdot 69+62.25\cdot 62+65\cdot 65+69.5\cdot 70+68\cdot 67+69\cdot 67+62\cdot 63+60.25\cdot 62=34462\\ {\displaystyle \sum x_i}=68+62.25+65\cdot 65+69.5+68+69+62+60.25=524\\ {{\displaystyle \sum x_i}}^2={68}^2+{62.25}^2+65\cdot {65}^2+{69.5}^2+{68}^2+{69}^2+{62}^2+{60.25}^2=34413.375\\ {\displaystyle \sum y_i=69+62+65+70+67+67+63+62=}525\\ {\displaystyle \sum y_i{}^2={69}^2+{62}^2+{65}^2+{70}^2+{67}^2+{67}^2+{63}^2+{62}^2=}34521\end{array}$

Find the sample variance $s^2$ using the formula $s^2=\frac{{\displaystyle \sum x_i{}^2}-\frac{{\displaystyle \sum x_i{}^2}}{n}}{n-1}$

Where $n=$ number of ordered pairs.

  $n=8$

  $\begin{array}{l}{s}_x{}^2=\frac{34413.375-\frac{{524}^2}{8}}{8-1}=\frac{34413.375-\frac{{524}^2}{8}}{7}\approx 13.05\\ s_y{}^2=\frac{34521-\frac{{525}^2}{8}}{8-1}=\frac{34521-\frac{{525}^2}{8}}{7}\approx 9.696\end{array}$

Find sample standard deviation.

  $\begin{array}{l}{s}_x=\sqrt{13.05}\approx 3.6\\ s_y=\sqrt{9.696}\approx 3.114\end{array}$

For covariance $s_{xy}$ using formula $s_{xy}=\frac{{\displaystyle \sum x_i{y}_i}-\frac{{\displaystyle \sum x_i{\displaystyle \sum y_i}}}{n}}{n-1}$ .

  $\begin{array}{l}{s}_{xy}=\frac{34462-\frac{524\cdot 525}{8}}{8-1}\\ =\frac{34462-\frac{524\cdot 525}{8}}{7}\\ \approx 10.643\end{array}$

Find the correlation coefficient $r$ using the formula $r=\frac{s_{xy}}{s_x{s}_y}$ .

  $r=\frac{10.643}{3.6\cdot 3.114}\approx 0.95$

Hence the correlation coefficient is $r\approx 0.95$ .

To determine

Find the slope of the line.

Answer to Problem 3.35SE

The slope of the line is $b\approx 0.815$ .

Explanation of Solution

The table shows the relation between eight people’s armspan and height.

  

Calculation:

First determine ${\displaystyle \sum x_i}{y}_i$ , ${\displaystyle \sum x_i}$ and ${\displaystyle \sum y_i}$ .

  $x_i\to$ first coordinates of the ordered pairs.

  $y_i\to$ second coordinates of the ordered pairs.

  $\begin{array}{l}{\displaystyle \sum x_i}{y}_i=68\cdot 69+62.25\cdot 62+65\cdot 65+69.5\cdot 70+68\cdot 67+69\cdot 67+62\cdot 63+60.25\cdot 62=34462\\ {\displaystyle \sum x_i}=68+62.25+65\cdot 65+69.5+68+69+62+60.25=524\\ {{\displaystyle \sum x_i}}^2={68}^2+{62.25}^2+65\cdot {65}^2+{69.5}^2+{68}^2+{69}^2+{62}^2+{60.25}^2=34413.375\\ {\displaystyle \sum y_i=69+62+65+70+67+67+63+62=}525\\ {\displaystyle \sum y_i{}^2={69}^2+{62}^2+{65}^2+{70}^2+{67}^2+{67}^2+{63}^2+{62}^2=}34521\end{array}$

Find the sample variance $s^2$ using the formula $s^2=\frac{{\displaystyle \sum x_i{}^2}-\frac{{\displaystyle \sum x_i{}^2}}{n}}{n-1}$

Where $n=$ number of ordered pairs.

  $n=8$

  $\begin{array}{l}{s}_x{}^2=\frac{34413.375-\frac{{524}^2}{8}}{8-1}=\frac{34413.375-\frac{{524}^2}{8}}{7}\approx 13.05\\ s_y{}^2=\frac{34521-\frac{{525}^2}{8}}{8-1}=\frac{34521-\frac{{525}^2}{8}}{7}\approx 9.696\end{array}$

Find sample standard deviation.

  $\begin{array}{l}{s}_x=\sqrt{13.05}\approx 3.6\\ s_y=\sqrt{9.696}\approx 3.114\end{array}$

For covariance $s_{xy}$ using formula $s_{xy}=\frac{{\displaystyle \sum x_i{y}_i}-\frac{{\displaystyle \sum x_i{\displaystyle \sum y_i}}}{n}}{n-1}$ .

  $\begin{array}{l}{s}_{xy}=\frac{34462-\frac{524\cdot 525}{8}}{8-1}\\ =\frac{34462-\frac{524\cdot 525}{8}}{7}\\ \approx 10.643\end{array}$

Find the correlation coefficient $r$ using the formula $r=\frac{s_{xy}}{s_x{s}_y}$ .

  $r=\frac{10.643}{3.6\cdot 3.114}\approx 0.95$

Find the slope $b$ using the formula $b=r\cdot \frac{s_y}{s_x}$

  $b=0.95\cdot \frac{3.114}{3.6}\approx 0.815$

Hence the slope of the line is $b\approx 0.815$ .

To determine

Find the regression line using the given points.

Answer to Problem 3.35SE

The regression line is $y=12.22+0.815x$ .

Explanation of Solution

The table shows the relation between eight people’s armspan and height.

  

Calculation:

First determine ${\displaystyle \sum x_i}{y}_i$ , ${\displaystyle \sum x_i}$ and ${\displaystyle \sum y_i}$ .

  $x_i\to$ first coordinates of the ordered pairs.

  $y_i\to$ second coordinates of the ordered pairs.

  $\begin{array}{l}{\displaystyle \sum x_i}{y}_i=68\cdot 69+62.25\cdot 62+65\cdot 65+69.5\cdot 70+68\cdot 67+69\cdot 67+62\cdot 63+60.25\cdot 62=34462\\ {\displaystyle \sum x_i}=68+62.25+65\cdot 65+69.5+68+69+62+60.25=524\\ {{\displaystyle \sum x_i}}^2={68}^2+{62.25}^2+65\cdot {65}^2+{69.5}^2+{68}^2+{69}^2+{62}^2+{60.25}^2=34413.375\\ {\displaystyle \sum y_i=69+62+65+70+67+67+63+62=}525\\ {\displaystyle \sum y_i{}^2={69}^2+{62}^2+{65}^2+{70}^2+{67}^2+{67}^2+{63}^2+{62}^2=}34521\end{array}$

Find the sample variance $s^2$ using the formula $s^2=\frac{{\displaystyle \sum x_i{}^2}-\frac{{\displaystyle \sum x_i{}^2}}{n}}{n-1}$

Where $n=$ number of ordered pairs.

  $n=8$

  $\begin{array}{l}{s}_x{}^2=\frac{34413.375-\frac{{524}^2}{8}}{8-1}=\frac{34413.375-\frac{{524}^2}{8}}{7}\approx 13.05\\ s_y{}^2=\frac{34521-\frac{{525}^2}{8}}{8-1}=\frac{34521-\frac{{525}^2}{8}}{7}\approx 9.696\end{array}$

Find sample standard deviation.

  $\begin{array}{l}{s}_x=\sqrt{13.05}\approx 3.6\\ s_y=\sqrt{9.696}\approx 3.114\end{array}$

For covariance $s_{xy}$ using formula $s_{xy}=\frac{{\displaystyle \sum x_i{y}_i}-\frac{{\displaystyle \sum x_i{\displaystyle \sum y_i}}}{n}}{n-1}$ .

  $\begin{array}{l}{s}_{xy}=\frac{34462-\frac{524\cdot 525}{8}}{8-1}\\ =\frac{34462-\frac{524\cdot 525}{8}}{7}\\ \approx 10.643\end{array}$

Find the correlation coefficient $r$ using the formula $r=\frac{s_{xy}}{s_x{s}_y}$ .

  $r=\frac{10.643}{3.6\cdot 3.114}\approx 0.95$

Find the slope $b$ using the formula $b=r\cdot \frac{s_y}{s_x}$

  $b=0.95\cdot \frac{3.114}{3.6}\approx 0.815$

Find the value of y-intercept.

  $\begin{array}{l}a=\overline{y}-b\overline{x}=\frac{{\displaystyle \sum y_i}}{n}-b\frac{{\displaystyle \sum x_i}}{n}\\ =\frac{525}{8}-\left(0.815\right)\frac{524}{8}\approx 12.22\end{array}$

Regression line is $y=a+bx$

  $y=12.22+0.815x$

Hence the regression line is $y=12.22+0.815x$ .

To determine

Find the height of the person for given value of armspan.

Answer to Problem 3.35SE

The person’s height is $62.77$ inches.

Explanation of Solution

The given value of armspan is $62$ .

Calculation:

The regression line is $y=12.22+0.815x$ .

Substitute $x=62$ in the given equation.

  $y=12.22+0.815\cdot \left(62\right)=62.77$

Hence the person’s height is $62.77$ inches.