9. The vice president of purchasing for a large national retailer has asked you to preparean analysi

9. The vice president of purchasing for a large national retailer has asked you to preparean analysis of retail sales by state. Data are available for the following variables: Y (retsal) = Per capita retail sales in $ X1 (perinc) = Per capita personal income in $ X2 (unempl) = Unemployment rate in % X3 (totpop) = State population in 000sExcel regression output of a potential model is:SUMMARY OUTPUTRegression StatisticsMultiple R 0.673063624R Square 0.453014642Adjusted R Square 0.430223585Standard ErrorNNN-NN-NNNNObservations 50ANOVAdf SS MS F Significance FRegression 2 14931938.3(NNN) NNN-NNNN149 19.87686003 5.14537E-07Residual 47 18029332.53(NNN) NNN-NNNNTotal 49 32961270.82Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 3054.280348(NNN) NNN-NNNN4.216390382 0.000109151 1597.811292 4510.749404unempl -86.25168104 40.20459701 -2.14531888 0.037015057 -167.0884398 -5.414922307perinc 0.253683705 0.048149492 5.268668342 3.2101E-06 0.156872664 0.350494746(a) Comment on the e ects of unemployment and per capita personal income.(b) You think the prediction equation can be improved by adding state populationas an additional explanatory variable. You obtained the following output:SUMMARY OUTPUTRegression StatisticsMultiple R 0.687861707R Square 0.473153727Adjusted R Square 0.439525242Standard Error(NNN) NNN-NNNNObservations 50ANOVAdf SS MS F Significance FRegression 3 15595748.15(NNN) NNN-NNNN716 14.07002785 1.12578E-06Residual 46 17365522.67(NNN) NNN-NNNNTotal 49 32961270.82Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 2828.429295(NNN) NNN-NNNN3.832870736 0.000375307 1343.885177 4312.973414unempl -71.32832666 41.40025242 -1.722895936 0.091481858 -154.6148904 11.95823702perinc 0.272491364 0.049773611 5.474615138 1.66335E-06 0.172359776 0.372622951totpop -0.024730373 0.018450316 -1.340376626 0.186566538 -0.061847621 0.0123868754i. Is this model better? Why/why not?ii. For this model, write out an expression for sales.iii. For this model, calculate a 95% con dence interval for predicted sales,if unemployment is 8.1%, per capita income is $15,000 and the state’spopulation is 6 million. Use a z-value of 1.96.(c) Write down two additional explanatory variables which you think could help toexplain sales. Give a brief justi cation for each.(25 marks)10. (a) Time series are usually considered to have a combination of four components.What are these components? For each of them, give one example of data forwhich you would expect that component to be present.(b) The following table gives average UK household electricity demand in kilowatthours (kWh) over the last ve years. Quarter 1 represents Spring.Year Q1 Q2 Q3 Q42005 4.5 4.1 4.4 5.12006 4.9 4.6 4.6 5.32007 5.0 4.7 4.8 5.52008 5.2 5.0 5.1 5.62009 5.3 5.1 5.2 5.7i. State two features about household electricity demand that are apparentfrom these data.ii. Show that the 4-point centred moving average for Quarter 3 in 2007 is5.025.iii. Calculate the ratio-to-moving-average (R2MA) for Quarter 3 in 2007.iv. Compute the four seasonal indices using the following table of R2MAvalues. Replace `?’ with your answer to part `iii.’.Season Autumn Winter Spring Summer0.961749 1.088000 1.026178 0.9533680.946015 1.084399 1.015228 0.944724? 1.081081 1.007264 0.9592330.973747 1.064133 1.002364 0.960000Seasonal averages * 1.079403 1.012759 0.954331v. The estimated trend line is found to be:^y = 4:461 + 0:050x;where x is the Quarter number (Q1 of 2005 corresponds to x = 1). Providea forecast, to three decimal places, for average UK household electricity5demand for the summer of 2015. Do you have any comment to make aboutthis forecast?

 

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