Chapter 1, Problem 12P

(a)

To determine

The shoes size and height of people in classroom by surveying and correlation.

Answer to Problem 12P

The shoes size and height of people height is linear.

Explanation of Solution

The data shoes size and height of people by surveying in the class room is,

Shoe size	Height
9.5	71
9.5	70
10.5	68
10.5	71
11	72
7.5	71
8.5	68
8.5	71
9.5	71

Table (1)

The scatter plot of above data is shown below from Table (1).

Figure (1)

From Figure (1), the life height and shoes size are shown on coordinate and abscissa. The plot is represented by Cartesian coordinate with the independent variable x on the horizontal axis and dependent variable y on the vertical axis. The Figure (1) gives the linear equation and linear line. So, the above analysis shows that there is a linear correlation between shoes size and height.

Thus, the shoes size and height of people is linear.

(b)

To determine

The correlation coefficient.

Answer to Problem 12P

The correlation coefficient is $7.12$.

Explanation of Solution

The data shoes size and height of people by surveying in the class room is,

Shoe size	Height
9.5	71
9.5	70
10.5	68
10.5	71
11	72
7.5	71
8.5	68
8.5	71
9.5	71

To evaluate the above data in the form of $x$ and $y$.

$x$	$y$	$xy$	$x^2$	$y^2$
9.5	71	674.5	90.25	5041
9.5	70	66.5	90.25	4900
10.5	68	714	110.25	4624
10.5	71	745.5	110.25	5041
11	72	792	121	5184
7.5	71	532.5	56.25	5041
8.5	68	578	72.25	4624
8.5	71	603.5	72.25	5041
9.5	71	764.5	90.25	5041
${\displaystyle \sum x}=85$	${\displaystyle \sum y}=633$	${\displaystyle \sum xy}=5471$	${\displaystyle \sum x^2}=813$	${\displaystyle \sum y^2=}44537$

Where,

• $x$ is the size of the shoes.
• $y$ is the height of people.

Formula to calculate correlation coefficient is,

$C=\frac{{\displaystyle \sum xy}-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{\sqrt{n\left({\displaystyle \sum x^2}\right)-{\left({\displaystyle \sum x}\right)}^2}\sqrt{n\left({\displaystyle \sum x^2}\right)-{\left({\displaystyle \sum x}\right)}^2}}$

Where,

• $C$ is the correlation coefficient.
• $n$ is the number of persons.

Substitute $5471$ for ${\displaystyle \sum xy}$, $633$ for ${\displaystyle \sum y}$, $813$ for ${\displaystyle \sum x^2}$, $44537$ for ${\displaystyle \sum y^2}$ and $85$ for ${\displaystyle \sum x}$ to find the $C$.

$\begin{array}{c}C=\frac{9\times 5471-85\times 633}{\sqrt{9\left(813\right)-{\left(85\right)}^2}\sqrt{9\left(44537\right)-{\left(633\right)}^2}}\\ =\frac{820.5}{115.09}\\ =7.12\end{array}$

Thus, the correlation coefficient is $7.12$.