Quiz 4: Estimating Demand

Business

The data below shows the per capita can consumption of soft drink per year related to price of six packs, per capita income, and mean temperature across the 48 adjoining states in the country US. img Calculate average cans per capita across 48 contiguous states in the country US by entering the formula img . Calculate average 6 - pack price across 48 contiguous states in the country US by entering the formula img . Calculate average income per capita across 48 contiguous states in the country US by entering the formula img . Calculate average mean temperature across 48 contiguous states in the country US by entering the formula img . To run regression, select the "Data" tab and click "Data Analysis" in the "Analysis" grouping. Input the data of dependent variable in the "Input Y-Range" field, and then enter data of independent variables in the "Input X -Range" field for multiple columns. Click "OK" to create the analysis as below. img The intercept is calculated as 514.267. The coefficients of 6-pack price, income per capita and mean temperature are - 242.97 , 1.22 and 2.93. Thus, the demand equation estimated using OLS is img .

a) The regression equation of the Sherwin-Williams Company's sales img on its selling price img is given as: img Where, img and img The company's average sales img are calculated as: img The average selling price img is calculated as: img Following calculations are done on the basis of the information given above. img For the above calculated values, img And, img Thus, the regression equation of the company's sales img on its selling price img is img . b) The value of intercept indicates that when the selling price is zero, the sales by Sherwin-Williams Company are as high as 390,371 gallons. The coefficient of img is img indicating that a dollar increase in selling price is expected to decrease sales by img in a given sales region. c) We want to test the hypothesis that whether prices are useful in predicting paint sales at 0.05 level of significance. That is, we would perform a statistical test to determine whether the sample value img is significantly different from zero. The null hypothesis is img , no relationship between sales and price. And, the alternative hypothesis is img , no relationship between sales and price. The standard deviation of error term is img Substitute appropriate values form table. img The standard deviation for the sample distribution is img At 5 percent level of significance, img -statistic can be calculated as img With 10 observations, a img -distribution for the sample statistic img will have img degrees of freedom. From Table 2 of Appendix B, the img -value is obtained as img The calculated value of t from is less than img , we reject null hypothesis img . Thus, we conclude that a linear and negative relationship exists between selling price and paint sales at the 5 percent level of significance. d) The coefficient of determination measures the proportion of total variation in dependent variable explained by the independent variable. That is, img . Following table shows the calculation of explained sum of square img , unexplained sum of square img and total sum of square img . img img Substitute appropriate values from table in img img Thus, the coefficient of determination is img . It implies that price as an independent variable explains 75 percent of the variation in paint sales. e) Following are the results of an analysis of variance (ANOVA) on the regression. img The null hypothesis is img (no relationship between sales and price) And, the alternative hypothesis is img (relationship exists between sales and price) With 10 observations and 1 independent variable, a img -distribution for each level of statistical significance has img degrees of freedom for numerator and img degrees of freedom for denominator. ( img is the total number of observations and img is the number of estimated regression parameters, including the constant term) From Table 3 of Appendix B, the img -ratio for the above mentioned degrees of freedom at the 5 percent level of significance is 5.32. The img statistic, however, in the regression model is calculated as 24.034. This value is quite greater than the critical value. Thus, we reject the null hypothesis of no relationship between the paint sales and selling price. f) Substitute img in the estimated regression equation to obtain the sales img forecast. img The best estimate for paint sales in a region, therefore, is img . The standard error of the regression img as calculated in part (c) is 16.43. Thus, an approximate 95 percent forecast interval of img is img Thus, we are 95 percent confident that the true sales of paint at price of $14.50 per gallon will be img . g) The price elasticity of demand img measures the responsiveness of change in quantity demanded of paint img to the change in its price img . It is calculated as img …… (1) where img is the coefficient of sales with respect to price. Substitute img , img and img in (1) img The point price elasticity of paint sales is img .

The data below shows the per capita can consumption of soft drink per year related to price of six packs, per capita income, and mean temperature across the 48 adjoining states in the U.S. img • Calculate average cans per capita across 48 contiguous states in the U.S. by entering the formula img in cell B52. • Calculate average 6-pack price across 48 contiguous states in the U.S. by entering the formula img in cell C52. The intercept is calculated as 514.267. The coefficients of 6-pack price, income per capita and mean temperature are img , 1.22 and 2.93. Thus, the demand equation estimated using OLS is img . According to the estimated regression equation: • If 6-pack price rises by one unit, the demand for soft drink cans per capita declines by 243 units (holding income and mean temperature constant). • If the per capita income of the increases by one unit, the demand for soft drink cans per capita increases by 1.22 units (holding 6-pack price and mean temperature constant). • Further, if mean temperature increases by one unit, the demand for soft drink cans per capita increases by 2.93 units (holding 6-pack price and income constant). The price elasticity of demand img measures the responsiveness of change in quantity demanded to change in price. It is calculated as img …… (1) where, img is the coefficient of 6-pack price for the estimated demand equation, img is the average demand and img is the average price. Substitute img , img and img in (1) img Thus, the price elasticity of soft drink demand is img .