5.1 The data below (data set fancy) concern the monthly sales figures of a shop which opened in January 1987 and sells gifts, souvenirs, and novelties. The shop is situated on the wharf at a beach resort town in Queensland, Australia. The sales volume varies with the seasonal population of tourists. There is a large influx of visitors to the town at Christmas and for the local surfing festival, held every March since 1988. Over time, the shop has expanded its premises, range of products, and staff.

1987 1988 1989 1990 1991 Jan 1664.81 2499.81 4717.02 5921.10 4826.64 Feb 2397.53 5198.24 5702.63 5814.58 6470.23 Mar 2840.71 7225.14 9957.58 12421.25 9638.77 Apr 3547.29 4806.03 5304.78 6369.77 8821.17 May 3752.96 5900.88 6492.43 7609.12 8722.37 Jun 3714.74 4951.34 6630.80 7224.75 10209.48 Jul 4349.61 6179.12 7349.62 8121.22 11276.55 Aug 3566.34 4752.15 8176.62 7979.25 12552.22 Sep 5021.82 5496.43 8573.17 8093.06 11637.39 Oct 6423.48 5835.10 9690.50 8476.70 13606.89 Nov 7600.60 12600.08 15151.84 17914.66 21822.11 Dec 19756.21 28541.72 34061.01 30114.41 45060.69

(a) Produce a time plot of the data and describe the patterns in the graph. Identify any unusual or unexpected fluctuations in the time series.

(b) Explain why it is necessary to take logarithms of these data before fitting a model.

(c) Use R to fit a regression model to the logarithms of these sales data with a linear trend, seasonal dummies and a "surfing festival"

dummy variable.

(d) Plot the residuals against time and against the fitted values. Do these plots reveal any problems with the model?

(e) Do boxplots of the residuals for each month.

Does this reveal any problems with the model?

(f) What do the values of the coefficients tell you about each variable?

(g) W hat does the Durbin-Watson statistic tell you about your model?
(h) Regardless of your answers to the above questions, use your regression model to predict the monthly sales for 1994, 1995, and 1996.
Produce prediction intervals for each of your forecasts.
(i) Transform your predictions and intervals to obtain predictions and intervals for the raw data.
(j) How could you improve these predictions by modifying the model?