Module $2$

Forecasting and decision analysis

Homework assignment $-$ HW$1$

$1$ Suppose you are tested to see if you have a rare disease. Assume that if you have the disease, your test will always come back positive, but if you do not have the disease, there is still a $0\mathrm{.}001$ chance that you will test positive. In reality, only $1$ of $10,000$ people have the disease. Your doctor calls and says that you have tested positive. He is sorry but there is a $99\mathrm{.}9\%(1-0\mathrm{.}001)$ chance that you have the disease. Is he correct? What is the actual probability that you have the disease?

$2$ Carlisle Tire and Rubber, Inc. is considering expanding production to meet potential increases in the demand for one of its tire products. Carlisle$$s alternatives are to construct a new plant, expand the existing plant, or do nothing in the short run. The market for this particular tire product may expand, remain stable, or contract. Carlisle$$s marketing department estimates the probabilities of these market outcomes to be $0\mathrm{.}35,0\mathrm{.}35,$ and $0\mathrm{.}30,$ respectively. The Image below shows Carlisle$$s payoffs and costs for the various combinations of decisions and outcomes $($you$$ll find this information in HW$1$ Problem $2).$

a$)$ Use decision tree to identify the strategy that maximizes the tire manufacturer$$s expected profit.

b$)$ Perform a sensitivity analysis on the optimal decision, letting each of the monetary inputs vary one at a time plus or minus $10\%$ from its base value, and summarize your findings. Which of the inputs appears to have the largest effect on the best solution?

$3$ The Worksheet HW$1$ Problem $3$ contains the monthly number of airline tickets sold by a travel agency.

a$)$ Does a linear trend appear to fit these data well? Given R$2,$ how can you obtain multiple R$?$ What do these numbers tell you about the linear trend?

b$)$ Is there evidence of some seasonal pattern in these sales data? If so$,$ characterize the seasonal pattern.

c$)$ In the case there is a seasonal pattern of this set of data, what would be a better method for you to use to forecast? Provide explanation to your answer.

$4$ The Worksheet HW$1$ Problem $4$ contains the monthly retail sales of U$.$S$.$ liquor stores.

a$)$ Is seasonality present in these data? If so$,$ characterize the seasonality pattern.

b$)$ Use Winters$$ method to forecast this series with smoothing constants $\backslash$alpha $=\backslash$beta $=0\mathrm{.}1$ and $\backslash$gamma $=0\mathrm{.}3.$ Does the forecast seem to track the seasonal pattern well? What are your forecasts for the next $5$ months?