Question

1 Problem 8 (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 belief, the following data was collected: Salesperson Number of Contacts Sales (in thousands) 14 24 2 12 14 3 20 28 4 16 30 5 46 80 6 23 30 7 48 90 8 50 85 55 120 10 50 110 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) e) Determine the least squares line. Provide an interpretation of your results. (8 marks)

Answer 1

х Y X*Y X2 y2 14 24 336 196 576 12 14 168 144 196 20 28 560 400 784 16 30 480 256 900 46 80 3680 2116 6400 23 30 690 529 900 48 90 4320 2304 8100 50 85 4250 2500 7225 55 120 6600 3025 14400 50 110 5500 2500 12100 Sum = 334 611 26584 13970 51581 The correlation coefficient r is computed using the following expression: r SSXY SSXX SSyy where 1 SSXY = 3xx -- (2x) () 2 1 SSxx = x X n i=1 i=1 2 72 SSyy = ΣΥ? In this case, based on the data provided, we get that SSXY = 26584 - (334 x 611) = 6176.6 10 1 SSXX = 13970 (334)2 = 2814.4 10 SSyy = 51581 (611)2 = = 14248.9 10 Therefore, based on this information, the sample correlation coefficient is computed as follows = SSXY SSXX SSyy 6176.6 V2814.4 x 14248.9 = 0.975

There exist a strong positive linear relationship between the two variables.

b.

Obs х Y X Y? X, Y, 1 14 24 196 576 336 2 12 14 144 196 168 3 20 28 400 784 560 4 16 30 256 900 480 5 46 80 2116 6400 3680 6 23 30 529 900 690 7 48 90 2304 8100 4320 8 50 85 2500 7225 4250 9 55 120 3025 14400 6600 10 50 110 2500 12100 5500 Sum = 334 611 13970 51581 26584 Based on the table above, we compute the following sum of squares that will be used in the calculation of the correlation coefficient 2 TE TI 1 SSxx x? 22 13970 1 10 x 111556 2814.4 sw - ☺ - (3) 1 = 51581 x 373321 10 14248.9 71 1 SSXY Šxx-! (Ex) CY 11 26584 x 204074 10 6176.6 Now, the correlation coefficientis computed using the following expression: SSXY SSxx SSyy 6176.6 2814.4 x 14248.9 0.9754 Then, the coefficient of determination, or R-Squared coefficient (R), is computed by simply squaring the correlation coefficient that was found above. So we get R2 = 0.97542 = 0.9513 Therefore, based on the sample data provided, it is found that the coefficient of determination is R² = 0.9513. This implies that approximately 95.13% of variation in the dependent variable is explained by the independent variable.

c.

х Y X*Y X2 y2 14 24 336 196 576 12 14 168 144 196 20 28 560 400 784 16 30 480 256 900 46 80 3680 2116 6400 23 30 690 529 900 48 90 4320 2304 8100 50 85 4250 2500 7225 55 120 6600 3025 14400 50 110 5500 2500 12100 Sum = 334 611 26584 13970 51581 Based on the above table, the following is calculated: 1 X = X; 334 10 = 33.4 n 2=1 n 611 Y = Y = 61.1 n 10 i=1 1 98xx = Î x? - : (2x) = 13970 – 3342/10 = 2814.4 2 n n 1 SSYY Y = 51581 – 6112/10 = 14248.9 n i=1 12=1 1 SSxy = x;Y; - -¿xx -- (_) () 26584 - 334 x 611/10 = 6176.6 Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n) are obtained as follows: m = SSXY SSxx 6176.6 2814.4 = 2.1946 n=Y - X m = 61.1 - 33.4 x 2.1946 = -12.201 Therefore, we find that the regression equation is: Y = -12.201 + 2.1946 X

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