Question
Estimate the regression equation of sales on advertising.
Answer :
Word Count : 997
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# Estimate the Regression Equation of Sales on Advertising Regression analysis is an important statistical technique used in business mathematics and statistics to study the relationship between two or more variables and to make estimates or predictions. In business decisions, sales are often influenced by advertising expenditure. A firm may spend money on newspapers, television, radio, social media, digital platforms, or other promotional activities with the expectation that higher advertising expenditure will increase sales. The regression equation of sales on advertising helps a business estimate the expected level of sales corresponding to a given amount of advertising expenditure. When sales and advertising are represented by variables, sales is generally treated as the dependent variable, while advertising expenditure is treated as the independent variable. Let (X) denote advertising expenditure and (Y) denote sales. The regression equation of sales on advertising is expressed in the form: [ Y=a+bX ] Here, (Y) represents the estimated or expected sales, (X) represents advertising expenditure, (a) is the intercept or constant, and (b) is the regression coefficient of sales on advertising. The value of (b) indicates the average change in sales resulting from a one-unit change in advertising expenditure. The regression equation is estimated from observed data relating advertising expenditure and sales. Suppose a business has information on advertising expenditure and corresponding sales for several years or periods. The values of (a) and (b) can be calculated by using the least-squares method. This method determines the line that gives the best fit to the observed data by minimizing the sum of the squared differences between actual sales and estimated sales. The regression equation can conveniently be estimated by using deviations from the arithmetic means. If (\bar X) is the mean advertising expenditure and (\bar Y) is the mean sales, the regression equation of (Y) on (X) is: [ Y-\bar Y=b_{yx}(X-\bar X) ] where (b_{yx}) is the regression coefficient of (Y) on (X). The regression coefficient is calculated as: [ b_{yx}=\frac{\sum (X-\bar X)(Y-\bar Y)} {\sum (X-\bar X)^2} ] Therefore, the estimated regression equation becomes: [ Y-\bar
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