Question

The coefficients of correlation between the heights (in inches) of fathers and sons from the following data will be: Heights of fathers $\left(x\right)65666768697071$ and Heights of Sons $\left(y\right)67686669727269$

• A. $0.60$
• B. $-0.60$
• C. $-0.67$
• D. $0.67$

Answer 1

The correct option is D

$0.67$

Explanation for the correct option:

Step 1: Finding mean of height of fathers and sons

Given heights of fathers as $x=\left\{65,66,67,68,69,70,71\right\}$ and their mean will be

$$\begin{gathered} \stackrel{-}{x}=\frac{65+66+67+68+69+70+71}{7} \\ \stackrel{-}{x}=68 \end{gathered}$$

Given heights of sons as $y=\left\{67,68,66,69,72,72,69\right\}$ and their mean will be

$$\begin{gathered} \stackrel{-}{y}=\frac{67+68+66+69+72+72+69}{7} \\ \stackrel{-}{y}=69 \end{gathered}$$

Step 2: Forming table of required terms

Now, creating the table with the required values to calculate the coefficient of correlation

$x-\stackrel{-}{x}$	$y-\stackrel{-}{y}$	$\left(x-\stackrel{-}{x}\right)\left(y-\stackrel{-}{y}\right)$	${\left(x-\stackrel{-}{x}\right)}^2$	${\left(y-\stackrel{-}{y}\right)}^2$
$-3$	$-2$	$6$	$9$	$4$
$-2$	$-1$	$2$	$4$	$1$
$-1$	$-3$	$3$	$1$	$9$
$0$	$0$	$0$	$0$	$0$
$1$	$3$	$3$	$1$	$9$
$2$	$3$	$6$	$4$	$9$
$3$	$0$	$0$	$9$	$0$
		$\sum _{}^{}\left(x-\overline{x}\right)\left(y-\overline{y}\right)=20$	$\sum _{}^{}{\left(x-\overline{x}\right)}^2=28$	$\sum _{}^{}{\left(y-\overline{y}\right)}^2=32$

Step 3: Calculating the coefficient of correlation

The coefficient of correlation is given by

$r=\frac{\sum _{}\left(x-\overline{x}\right)\left(y-\overline{y}\right)}{\sqrt{\sum _{}{\left(x-\overline{x}\right)}^2\sum _{}{\left(y-\overline{y}\right)}^2}}$

On substituting the values we get,

$$\begin{gathered} r=\frac{20}{\sqrt{28\times 32}} \\ r=0.67 \end{gathered}$$

Hence, option (D) is the correct answer.