$($Forecasting$)$ Weekly demand for a product is given in the table below:

$\backslash$table$[[$Week$,1,2,3,4,5,6,7,8,9,10],[\backslash$table$[[$Demand$],[($units$)]],24,25,27,26,29,31,32,32,35,37]]$

$($a$)$ Using Microsoft Excel, plot the weekly demands, and comment on it by considering the components of a time series.

$($b$)$ Apply the $5-$week moving average method to make forecasts for weeks $6$ through $10.$

$($c$)$ Take the estimated value of the level at the end of week $5$ as the moving average value computed in part $($b$)$ for week $5.$ Choose $=0\mathrm{.}2.$ Apply the method of simple exponential smoothing to determine single$-$week$-$ahead forecasts for demands $6$ through $10.$

$($d$)$ Choose $=0\mathrm{.}2$ and $=0\mathrm{.}3$ and apply the method of double exponential smoothing to determine single$-$demand$-$ahead forecasts for demands $6$ through $10.$

$($e$)$ Using Microsoft Excel, on the same graph, plot the forecast errors of the methods in parts $($b$),($c$),$ and $($d$).$ Do you observe any bias in the methods in parts $($b$),($c$),$ and $($d$)?$ Show your answer by any method of measuring bias.

$($f$)$ Consider the forecast errors of the method of double exponential smoothing. Compute $95\mathrm{.}5$ percent control limits for the forecast errors, assuming that the forecast errors are normally distributed. Using Microsoft Excel, draw a control chart for the forecast errors, and check whether the forecast errors are in control.

$($g$)$ Which of the forecasting methods in parts $($b$),($c$),$ and $($d$)$ do you select for implementation? Show your work.