Question

Let x be the average number of employees in a group health insurance plan, and let y be the average administrative cost as a percentage of claims.

x	3	7	15	38	70
y	40	35	30	26	16

(a) Make a scatter diagram of the data and visualize the line you think best fits the data.

(b) Would you say the correlation is low, moderate, or strong? positive or negative?

low and positive

moderate and positive     

moderate and negative

strong and positive

low and negative

strong and negative

(c) Use a calculator to verify that Σx = 133, Σx² = 6627, Σy = 147, Σy² = 4657, and Σxy = 2923. Compute r. (Round your answer to three decimal places.)
r =  

As x increases, does the value of r imply that y should tend to increase or decrease? Explain.

Given our value of r, y should tend to remain constant as x increases.

Given our value of r, y should tend to increase as x increases.     

Given our value of r, we cannot draw any conclusions for the behavior of y as x increases.

Given our value of r, y should tend to decrease as x increases.

Answer 1

a

.

a.

scatter plot shown above

Line of Regression Y on X i.e Y = bo + b1 X
					X	Y	(Xi - Mean)^2	(Yi - Mean)^2	(Xi-Mean)*(Yi-Mean)
3	40	556.96	112.36	-250.16
7	35	384.16	31.36	-109.76
15	30	134.56	0.36	-6.96
38	26	129.96	11.56	-38.76
70	16	1883.56	179.56	-581.56

calculation procedure for regression
mean of X = sum ( X / n ) = 26.6
mean of Y = sum ( Y / n ) = 29.4
sum ( (Xi - Mean)^2 ) = 3089.2
sum ( (Yi - Mean)^2 ) = 335.2
sum ( (Xi-Mean)*(Yi-Mean) ) = -987.2
b1 = sum ( (Xi-Mean)*(Yi-Mean) ) / sum ( (Xi - Mean)^2 )
= -987.2 / 3089.2
= -0.32
bo = sum ( Y / n ) - b1 * sum ( X / n )
bo = 29.4 - -0.32*26.6 = 37.9
value of regression equation is, Y = bo + b1 X
Y'=37.9-0.32* X          

b.

( X)	( Y)	X^2	Y^2	X*Y
3	40	9	1600	120
7	35	49	1225	245
15	30	225	900	450
38	26	1444	676	988
70	16	4900	256	1120

calculation procedure for correlation
sum of (x) = 133
sum of (y) = 147
sum of (x^2) = 6627
sum of (y^2) = 4657
sum of (x*y) = 2923
to calculate value of r( x,y) = co variance ( x,y ) / sd (x) * sd (y)
co variance ( x,y ) = [ sum (x*y - N *(sum (x/N) * (sum (y/N) ]/n-1
= 2923 - [ 5 * (133/5) * (147/5) ]/5- 1
= -197.44
and now to calculate r( x,y) = -197.44/ (SQRT(1/5*2923-(1/5*133)^2) ) * ( SQRT(1/5*2923-(1/5*147)^2)
=-197.44 / (24.856*8.188)
=-0.97
value of correlation is =-0.97
coefficient of determination = r^2 = 0.941
properties of correlation
1. If r = 1 Correlation is called Perfect Positive Correlation
2. If r = -1 Correlation is called Perfect Negative Correlation
3. If r = 0 Correlation is called Zero Correlation
& with above we conclude that correlation ( r ) is = -0.9701< 0, perfect negative correlation          

low and negative   

c.

value of correlation is =-0.97
coefficient of determination = r^2 = 0.941

As X increases then r ,y values also increases