Question

The manager of an insurance company wants to know the correlation number of customer services representatives and the number of returning insured each year. The results are as follows:

# of reps	# of returning insureds (100s)
1 2 3 5 8 11 11 17 20 25	5 10 15 17 18 20 20 26 27 30

Find the correlation between customer service representatives and returning insured. ??

Answer 1

Solution:

Given :

The manager of an insurance company wants to know the correlation number of customer services representatives and the number of returning insured each year. The results are as follows:

# of reps	# of returning insureds (100s)
1 2 3 5 8 11 11 17 20 25	5 10 15 17 18 20 20 26 27 30

Correlation coefficient formula is :

$r=\frac{SS_{xy}}{\sqrt{SS_{xx}\times SS_{yy}}}$

where

$SS_{xy}=\sum xy \: \: -\: \: \left ( \sum x\times \sum y \: \: /\: \: n\: \: \right )$

$SS_{xx}=\sum x^{2} \: \: -\: \: \left ( \sum x\times \sum x \: \: /\: \: n\: \: \right )$

$SS_{yy}=\sum y^{2} \: \: -\: \: \left ( \sum y\times \sum y \: \: /\: \: n\: \: \right )$

Thus we need to make following table:

x : # of reps	y : # of returning insureds (100s)	x^2	y^2	xy
1	5	1	25	5
2	10	4	100	20
3	15	9	225	45
5	17	25	289	85
8	18	64	324	144
11	20	121	400	220
11	20	121	400	220
17	26	289	676	442
20	27	400	729	540
25	30	625	900	750
$\sum x = 103$	$\sum y = 188$	$\sum x^{2} = 1659$	$\sum y^{2} = 4068$	$\sum xy= 2471$

Thus

$SS_{xy}=\sum xy \: \: -\: \: \left ( \sum x\times \sum y \: \: /\: \: n\: \: \right )$.

$SS_{xy}=2471 \: \: -\: \: \left ( 103\times 188 \: \: /\: \: 10\: \: \right )$

$SS_{xy}=534.6$

$SS_{xx}=\sum x^{2} \: \: -\: \: \left ( \sum x\times \sum x \: \: /\: \: n\: \: \right )$

$SS_{xx}=1659 \: \: -\: \: \left ( 103 \times 103 \: \: /\: \: 10\: \: \right )$

$SS_{xx}=598.1$

$SS_{yy}=\sum y^{2} \: \: -\: \: \left ( \sum y\times \sum y \: \: /\: \: n\: \: \right )$

$SS_{yy}=4068 \: \: -\: \: \left ( 188 \times 188 \: \: /\: \: 10\: \: \right )$

$SS_{yy}=533.6$

Thus correlation coefficient is :
$r=\frac{SS_{xy}}{\sqrt{SS_{xx}\times SS_{yy}}}$

$r=\frac{534.6 }{\sqrt{598.1 \times 533.6 }}$

$r=\frac{534.6 }{\sqrt{319146.16 }}$

$r=\frac{534.6 }{564.9302258 }$

$r=0.9463$

Thus the correlation between customer service representatives and returning insured is r = 0.9463 .