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1 1) The S & P/TSX Composite Index is based on common stock prices of a group of Canadian stocks. The weekly close level of the TSX for 6 weeks are shown: Week TSX Index a) Compute the three week moving averages and forecast the TSX Index forecast for week 7 b) Compute the three week weighted moving averages using 3, 2 and 1 for most recent, second most recent, and third most recent periods. Find the TSX Index forecast for week 7. c) Compute the exponentially smoothed forecasts for weeks 2 to 7 using α = 0.7. d) Calculate the MAD and MAPE for all three methods. Which provides the best forecast for the TSX Index? 2) The following data represents the quarterly piano sales of Sawyer Piano House for 3 consecutive years. Year Quarter Sales (000$) a) Compute the seasonal indices and normalized indices using overall average sales. b) Compute the seasonal indices and normalized indices using centred moving average sales. c) If the average quarterly forecast for year 4 is $10,000, use the seasonal indices (unnormalized) to calculate seasonally adjusted quarterly forecasts for year 4. 1

2 3) James Steven has been hired by the Victory Stores, a convenience store, to study how factors such as floor space area, number of parking spaces, and average family income of families in the city affect daily sales. A random sample of 15 stores is obtained and the data are as follows: Sales ($) Floor Area Parking Spaces Income ($ 000) SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 15 ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept E Floor Area Parking Spaces Income ($ 000)

3 a) From the Excel output above determine the regression equation, adjusted coefficient of determination, correlation coefficient, standard error of the estimate. Interpret the meaning of each. b) Interpret the coefficients of each of the independent variables. c) Forecast a sales level for 500 (floor area), 5 (parking spaces) and 50 (Income, 000s). d) Complete the coefficients table by filling in the missing data for t stat (Floor Area) and Lower and Upper 95% confidence limits (Parking Spaces). e) Conduct an overall hypothesis test to determine if the regression equation is significant (useful) in explaining differences in Sales. Use a 5% significance level. f) Conduct individual hypothesis tests to determine which independent variables are significant or should be dropped in explaining differences in Sales. Use a 5% significance level. 4) Complete the following ANOVA table, assuming the sample size is 20. ANOVA df SS MS F Regression Residual (Error) Total 3

4 Definitions and concepts to know 1. Time Series Forecasting 2. Causal Forecasting 3. Qualitative Forecasting (a) Delphi Method (b) Jury of Executive Opinion (c) Sales Force Composite (d) Consumer Market Survey 4. MAD 5. MAPE 6. Trend 7. Seasonal 8. Cyclical 9. Random 10. Moving Average 11. Weighted Moving Average 12. Exponential Smoothing, Smoothing constant or parameter 13. Stationary, Non-stationary 14. ANOVA Table 15. Simple Regression 16. Multiple Regression 17. Correlation Coefficient 18. Coefficient of Determination 19. Standard Error of the Estimate 4

5 Answers/Solutions 1) a) b) c) Error Actual Week Actual 3 Week Moving Error Error TSX Index Average Forecast Totals MAD = 21.7 MAPE = 0.257% Error Actual Week Actual 3 Week Moving Error Error TSX Index Average Forecast Totals MAD = 21.0 MAPE = 0.243% Error Actual Week Actual Exponentially Smoothed Error Error TSX Index Forecast α = (assumed) Totals MAD = 13.8 MAPE = 0.162% d) Exponential Smoothing provides the best forecast due lowest MAD (13.8) and MAPE (0.162%). 5

6 2) a) b) Year Quarter Sales Overall Seasonal Seasonal Seasonal Index Average Sales Ratios Index (Normalized) Total Total Seasonal Ratios Year Quarter 1 Quarter 2 Quarter 3 Quarter Average Seasonal Indices Year Quarter Sales Centred Moving Seasonal Seasonal Seasonal Index Average Sales Ratios Index (Normalized) Total Total Seasonal Ratios Year Quarter 1 Quarter 2 Quarter 3 Quarter Average Seasonal Indices 6

7 c) For Overall Average Year 4 Q1 = Q2 = 4860 Q3 = 5400 Q4 = For Centred Average Year 4 Q1 = Q2 = 5230 Q3 = 4210 Q4 = In the case of centred average, better to allocate an annual estimate of with the normalized indices. Year 4 Q1 = Q2 = 5480 Q3 = 4400 Q4 = ) a) From Summary Output Regression equation ŷ = x x x 3 Coefficient of Determination = R 2 = Correlation Coefficient = Standard Error of Estimate = ± Meanings - see Practice Q1 and Causal Forecasting Model notes b) For each additional square foot of floor area, we expect sales to increase by $, all else being held constant. For each additional parking place, we expect sales to increase by $, all else being held constant. For each additional ( 000)$ average income, we expect sales to decrease by $ c) $1781 d) t stat (Floor Area) = 4.479; Lower 95% (Parking Spaces) 4.269, Upper 95% (Parking Spaces) e) H 0 : β 1 = β 2 = β 3 = 0 H 1 : Not all β s are 0 F Crit = So we reject H 0 and conclude that the linear relationship exists and at least one of the regression coefficients is not zero. 7

8 f) For Floor Area: For Parking Spaces: For Income: H 0 : β 1 = 0 H 0 : β 2 = 0 H 0 : β 3 = 0 H 0 : β 1 0 H 0 : β 2 0 H 0 : β 3 0 Critical t value = (or ) Floor Area and Parking Spaces, we reject the null hypothesis and keep these two variables in the model. We fail to reject the null hypothesis and discard Income as an independent variable. 4) ANOVA df SS MS F Regression Residual (Error) Total

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