Question

A sales manager for an advertising agency believes there is a relationship between the number of contacts and the amount of the sales. To verify this, the following data was collected:
Sales ($)   Contacts
25,000   15
16,000   15
28,000   20
60,000   18
80,000   46
35,000   24
85,000   45
80,000   45
95,000   55
110,000   52
Which is the:
Independent variable?   ___________________

Dependent variable?   ___________________

Calculate the coefficient of correlation and interpret the results.

Determine both the coefficient of determination and non-determination and explain the result.

Determine the unrounded regression equation for this data set.

Using the regression equation predict the amount of sales when 50 contacts are made. Show all work involved in finding your answer.

Answer 1

Here independent variable is Number of contacts

Dependent variable is Sales (In thousands)

Here coefficient of correlation is calculated with the help of values of , , , ,

	Contacts(x)	Sales ($)(y)	x^2	xy	y^2
	15	25	225	375	625
	15	16	225	240	256
	20	28	400	560	784
	18	60	324	1080	3600
	46	80	2116	3680	6400
	24	35	576	840	1225
	45	85	2025	3825	7225
	45	80	2025	3600	6400
	55	95	3025	5225	9025
	52	110	2704	5720	12100
Sum	335	614	13645	25145	47640

Correaltion coefficient r = [n(Σxy) - (Σx)((Σy)] / sqrt [ (n (Σx2 ) - (Σx)2) ( n (Σy2 ) - (Σy)2)]

r = [10 * 25145 - 335 * 614]/ [(10 * 13645 - 335 * 335) * (10 * 47640 - 335 * 335)]

r = 0.9325

Here as the correlation coefficient is near to 1 that means there is strong positive correlation between number of contacts and sales.

Here let say the regression equation is

y^ = a + bx

a = [(Σy) (Σx2 ) - (Σx) (Σxy)]/ [ n (Σx2 ) - (Σx)2 ]

a = [614 * 13645 - 335 * 25145]/ [10 * 13645- 335 * 335]

a = -1.88

b = [ n(Σxy) - (Σx)((Σy)]/ [ n (Σx2 ) - (Σx)2 ]

b = [10 * 25145 - 335 * 614]/ [10 * 13645- 335 * 335]

b = 1.889

y^ = -1.88 + 1.889x

where x = number of contacts

y^ = predicted value of sales (in thousands dollars)

Here if x = 50

y^ = -1.88 + 1.889 * 50

y^ = $92568 is  the amount of sales when 50 contacts are made.