Question

SOLVED PROBLEM 4.7 SALES, y ADVERTISING, x ку Cengiz Haksever runs an Istanbul high-end jewelry shop. He advertises weekly in local Turkish newspapers and is thinking of increasing his ad budget. Before doing so, he decides to eval- uate the past effectiveness of these ads. Five weeks are sampled and the data are shown in the table below: 11 5 25 55 6 3 9 18 10 49 70 6 4 12 12 64 96 y45 x25 E151 Σν 251 AD BUDGET SALES ($1,000s) THAT WEEK ($100s) 45 = 9 25 = 5 11 6 3 251 (55)9 151 (5K5 Zху - иху 10 7 6 251 225 26 12 8 151 125 26 Develop a regression model to help Cengiz evaluate his advertising a b 9 (1(5)-4 So the regression model is f 4lx, or Sales (in S1,000s) 4+1 (Ad budget in S100s) This means that for each 1-unit increase inx (or $100 in ads) sales increase by 1 unit (or S1,000). SOLUTION We apply the least-squares regression model as we did in Example 12

Answer 1

We can find the coefficient of determination i.e. r^2 using the solution provided.

Sum of y^2 = 11^2 + 6^2 + 10^2 + 6^2 + 12^2

= 437

r = n*sum of x*y - sum of x*sum of y/[(n*sum of x^2 - (sum of x)^2)*( n*sum of y^2 - (sum ofy)^2)]^0.5

= 5*251 – 25*45/[(5*151 – 25^2)*(5*437 – 45^2)]^0.5

= 0.9014

Thus r^2 = 0.9014^2 = 0.8125.

This means that 81.25% of the variability in sales can be explained by the regression model with advertising as the independent variable.