Question $1(30$ points$)$: A logistics company tracks the monthly demand for a particular product over the past seven months, as shown below:

$\backslash$table$[[$Month$,$Demand $($Units$),$Forecast $($MA$(3)),$Error$],[1,130,-,-],[2,145,-,-],[3,150,-,-],[4,145,,],[5,180,,],[6,175,,],[7,195,,]]$

a$)(10$ points$)$ Using a $3-$month moving average, forecast the demand for the $4^{th},5^{th},6^{th},$ and $7^{th}$ months.

b$)(10$ points$)$ Calculate the error terms for months $4,5,6,$ and $7$ based on the actual demand values in the table and the forecast values you obtained in part $($a$).$ Calculate the Mean Absolute Deviation Error $($MAD$)$ and Mean Squared Error $($MSE$).$

c$)(10$ points$)$ Calculate the $2-$month and $4-$month moving averages for Month $7$ and compare the forecasts. Based on the actual demand of $195$ units for Month $7,$ which moving average method $(2-$month, $3-$month, or $4-$month$)$ provides the most accurate forecast and explain the reason why it gives a smaller error term?