$10\mathrm{.}25.$ 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 inaccurate 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$.$ Based on the MAD and MSE values. which of these three methods do you recommend using hereafter?