Chapter 4.1, Problem 20P

(a)

To determine

To explain: Whether the value of the correlation coefficient will change if the symbols for x and y are exchanged in the formula.

Answer to Problem 20P

Solution: The correlation coefficient will remain the same if the symbols for x and y are exchanged.

Explanation of Solution

Given: The formula for the correlation coefficient is given in the question as:

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

The numerator of the above formula is $n{\displaystyle \sum xy\text{-}\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}$ and, in this formula, ${\displaystyle \sum xy={\displaystyle \sum yx}}$ and $\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)=\left({\displaystyle \sum y}\right)\left({\displaystyle \sum x}\right)$. So, any change in the symbols of the variables will give the same correlation coefficient. The denominator of the above formula is,

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

In this part, the values are squared; so, ultimately, they will show a positive result and hence, it will not make any difference to the resultant value of r.

(b)

To determine

To explain: Whether the value of the correlation coefficient will change if the values of x and y are exchanged.

Answer to Problem 20P

Solution: The correlation coefficient remains the same if the corresponding values of x and y are exchanged.

Explanation of Solution

Given: A hypothetical set of x and y data values are given in the question. The formula for correlation coefficient is also provided in it as:

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

The numerator of the above formula is $n{\displaystyle \sum xy\text{-}\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}$ and, in this formula, ${\displaystyle \sum xy={\displaystyle \sum yx}}$ and $\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)=\left({\displaystyle \sum y}\right)\left({\displaystyle \sum x}\right)$. So, any change in the symbols of the variables will give the same correlation coefficient. The denominator of the above formula is,

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

In this part, the values are squared; so, ultimately, they will show a positive result and hence, it will not make any difference to the resultant value of r.

(c)

To determine

To find: The value of the correlation coefficient of the two samples, and to show that the computed value of r is the same for both samples.

Answer to Problem 20P

Solution: The computed value of the correlation coefficient of both data sets is 0.618.

Explanation of Solution

Given: Consider the first data set:

x	1	3	4
y	2	1	6

The second data set is provided below:

x	2	1	6
y	1	3	4

Calculation: The value of the correlation coefficient of data set 1 can be computed by using the formula given below:

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

To compute ${\displaystyle \sum x}$, ${\displaystyle \sum y}$, ${\displaystyle \sum x^2}$ $,{\displaystyle \sum y^2}$ and ${\displaystyle \sum xy}$, it is easy to organize the data into a table of five columns and add the values in each column. The table is given below:

x	y	$x^2$	$y^2$	$xy$
1	2	1	4	2
3	1	9	1	3
4	6	16	36	24
${\displaystyle \sum x=8}$	${\displaystyle \sum x=9}$	${\displaystyle \sum x^2=26}$	${\displaystyle \sum y^2=41}$	${\displaystyle \sum xy=29}$

Substitute the values of ${\displaystyle \sum x}$, ${\displaystyle \sum y}$, ${\displaystyle \sum x^2}$, ${\displaystyle \sum y^2}$ and ${\displaystyle \sum xy}$ in the above formula.

$\begin{array}{c}r=\frac{3\left(29\right)-\left(8\right)\left(9\right)}{\sqrt{3\left(26\right)-{\left(8\right)}^2}\sqrt{3\left(41\right)-{\left(9\right)}^2}}\\ =\frac{15}{2.44306}\\ =0.618\end{array}$

Now, the correlation coefficient of data set 2 can be calculated as:

x	y	$x^2$	$y^2$	$xy$
2	1	4	1	2
1	3	1	9	3
6	4	36	16	24
${\displaystyle \sum x=9}$	${\displaystyle \sum y=8}$	${\displaystyle \sum x^2=41}$	${\displaystyle \sum y^2=26}$	${\displaystyle \sum xy=29}$

Substitute the values of ${\displaystyle \sum x}$, ${\displaystyle \sum y}$, ${\displaystyle \sum x^2}$, ${\displaystyle \sum y^2}$ and ${\displaystyle \sum xy}$ in the formula.

$\begin{array}{c}r=\frac{3\left(29\right)-\left(9\right)\left(8\right)}{\sqrt{3\left(41\right)-{\left(9\right)}^2}\sqrt{3\left(26\right)-{\left(8\right)}^2}}\\ =\frac{15}{24.24}\\ =0.618\end{array}$

Interpretation: For the data set 1, the value of r is 0.618 and for the data set 2, the value of r is also 0.618. This means that the value of the correlation coefficient remains the same if the values of x and y are exchanged.