Question

1 Problem 7 (20 marks) 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 believe, the following data was collected: Salesperson Number of Contacts Sales (in thousands) 14 24 2 12 14 3 20 28 4 16 30 16 80 23 7 48 90 50 85 55 120 10 SO 110 6 8 9 Assume normality of variables. a) Calculate the coefficient of correlation r. Provide an interpretation of the computed value of r. (9 marks) b) Calculate the coefficient de determination r square (R2). Provide an interpretation of the computed value of r square (R2). (3 marks) c) Determine the least squares line. Provide an interpretation of your results. (8 marks) 5

Answer 1

Observe that amount of the sales can be dependent on the number of contacts. So,

Independent variable(X) : Number of contacts.

Dependent variable(Y) : Amount of sales(in thousands).

a) correlation coefficient r n ZXili - Ezidy; [n2x?- (2xci)2] [n 24 - (24;)?] - 1749; = (4x24) +(2714)+(20%28) + +(55*120) +(50X110) 26584 Isi 14+12+207 +55750 334 9, 24+14+28+ ---+ 120 +110 3.611. Σα = 142+12+2624 +5524502= 13970. 2 = 24²+14+ n = 10 ( sample size) # 120241202 = 51581 ... (10 x 26584)-(334x611) [8.ox 13970)-(834)2](620x 51582)-(629)] ro 0.9753621 : a Value of correlation coefficient near I indicates that there is strong positive linear relationship between two variables. r= 0.9753621 can be interpreted as There is a strong positive linear relationship between number of contacts and Sales (in thousands) le as the number of contandts increases, the sales also increases.

b) coefficient of determination (R2) is the square of the correlation coefficient. R?= (0.9753621)? = 0.9513313 coefficient of determination CRP) is 0.9513313, coefficient of determination is interpreted as the amount of variation in y explained by So, Variation in the data on sales is explained by the data on Number of contacts. 95.13313% of c) In general, least square line is ŷ F âo the di where Bo is intercept ; ße is slope i po = 9 - u 61'1 ệs = Exit; - näy Exi? - noe? x = Izi n Ø2 = 26584-(10x 3364 X61-1) 13970 10 x 33:42 33:44 : 4 = 23; = 2: 195 -12.201 Bo= - ß2 2 = 61.1 - ( 2.195 X33:4) = least square line is y;= -12-201 + 2.195 Ri

Slope of the least square line is interpreted as the the change in dependent variable y for every unit change in the independent variable x. Here slope is 2.195 and it has positive sign so it can be interpreted as amount of sales increases by 2.195 thousands when the number of contacts increases by one.

Intercept of the least square line is interpreted as the expected estimated value of dependent variable y when independent variable x assumes zero value. Here intercept is -12.201, which can be interpreted as expected estimated amount of sales is -12.201 when the number of contacts is zero.

Note that the interpretation of intercept in this context doesn't make sense because amount of sales cannot be negative.

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