$10\mathrm{.}25,$ The Andes Mining Company mines and ships copper ore. The company's sales

manager, Juanita Valdes, has been using the moving$-$average method based on the last three

years of sales to forecast the demand for the next year. However. she has become dissatisfied

with the inaceurate forecasts being provided by this method.

The annual demands $($in tons of copper ore$)$ over the past $10$ years are $382,405,398,421,$

$426,415,443,451,446,464.$

a$.$ Explain why this pattern of demands inevitably led to significant inaccuracies

in the moving$-$average forecasts:

E b$.$ Determine the moving$-$average forecasts for the past seven years. What are

MAD and MSE? What is the forecast for next year?

E c$.$ Determine what the forecasts would have been for the past $10$ years if the

exponential smoothing method had been used instead with an initial estimate

of $380$ and a smoothing constant of $=0\mathrm{.}5.$ What are MAD and MSE? What

is the forecast for next year?

E d$.$ Determine what the forecasts would have been for the past $10$ years if

exponential smoothing with trend had been used instead. Use initial estimates

of $370$ for the average value and $10$ for the trend, with smoothing constants

$=0\mathrm{.}25$ and $=0\mathrm{.}25.$

c$.$ Biased on the MAD and MSE values, which of these three methods do you

recommend using hereafter?