Question

JESTION 5 (20 Marks) The Managers for an insurance firm are interested in finding out if the number of new clients a broker brings into a firm alfects the sales generated by the broker. A sample of six brokers was selected and the data is presented below: BROKER NUMBER OF CLIENTS SALES (R 000's) 2 2 7 94 5 10 98 5.1 Draw a scatter diagram depicting the data. (5 marks) 5.2 Calculate the Pearson's correlation coefficient. (10 marks) 5.3 Determine the linear regression equation and use it to predict the amount of sales for a broker who brought in 42 new (5 marks) clients into a firm. END OF PAPER

Answer 1

5.1)Scatter Diagram

64- 62- 60- 58 56 54- 52 50- 48 sales (R 000's) 46- 44 42 40- 38 36- 25 30 35 15 number of clients 10 5

-------------------------------------------------------------------------------------------------------------------------------------------------------

-------------------------------------------------------------------------------------------------------------------------------------------------------

5.2) Pearson Correlation Coefficient (R) = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

Let X = number of clients

Let Y = sales (R 000's)

X Values
∑ = 115
Mean = 19.167
∑(X - M_x)² = SS_x = 712.833

Y Values
∑ = 285
Mean = 47.5
∑(Y - M_y)² = SS_y = 633.5

X and Y Combined
N = 6
∑(X - M_x)(Y - M_y) = 653.5

Now Pearson Correlation Coefficient (R) can be calculated as below
R = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

R = 653.5 / √((712.833)(633.5)) = 0.9725
-------------------------------------------------------------------------------------------------------------------------------------------------------

-------------------------------------------------------------------------------------------------------------------------------------------------------

5.3) Linear Regression Equation

Here sum of our X's (Number of clients) = 115

Hense mean of X's (Mx) = 19.1667

Now sum of our Y's (sales) = 285

Hense mean of Y's (My)= 47.5

	X-Mx	Y-My	(X-Mx)2	(X-Mx)(Y-My)
	2.8333	4.5	8.0278	12.75
	-13.1667	-10.5	173.3611	138.25
	17.8333	16.5	318.0278	294.25
	8.8333	7.5	78.0278	66.25
	-9.1667	-6.5	84.0278	59.5833
	-7.1667	-11.5	51.3611	82.4167
Sum			712.8333	653.5

Sum of squares (SS_X) = 712.8333
Sum of products (SP) = 653.5

Regression Equation = Y = bX + a

b=SP/SSx = 653.5/712.83 = 0.91676

a=M_Y - bM_{X =} 47.5 - (0.92*19.17) = 29.92869

Regression Equation = Y = bX + a = 0.91676X+29.92869