QUESTION $3$

Transcontinental Airlines maintains a computerized forecasting system to forecast the number of customers in each fare class who will fly on each flight in order to allocate the available reservations to fare classes properly. For example, consider economy$-$class customers flying in midweek on the noon flight from New York to Los Angeles. The following table shows the average number of such passengers during each month of the year just completed. The table also shows the seasonal factor that has been assigned to each month based on historical data. Show all formulas and calculations.

$\backslash$table$[[$Month$,\backslash$table$[[$Average$],[$Number$]],\backslash$table$[[$Seasonal$],[$Factor$]],$Month,$\backslash$table$[[$Average$],[$Number$]],\backslash$table$[[$Seasonal$],[$Factor$]]],[,,,,,],[$January$,70,0\mathrm{.}9,$July,$96,1\mathrm{.}17],[$February$,73,0\mathrm{.}88,$August,$98,1\mathrm{.}15],[$March$,68,0\mathrm{.}91,$September,$82,0\mathrm{.}97],[$April$,74,0\mathrm{.}93,$October,$75,0\mathrm{.}91]]$

PRDEAPS$/2024$

$\backslash$table$[[$May$,79,0\mathrm{.}96,$November,$86,1\mathrm{.}05],[$June$,87,1\mathrm{.}09,$December,$91,1\mathrm{.}08]]$

$($a$)$ After considering seasonal effects, compare the MAD values for the last$-$value method, the averaging method, the moving average method $($based on the most recent three months$),$ and the exponential smoothing method $($with an initial estimate of $80$ and a smoothing constant of $0\mathrm{.}2$ when they are applied retrospectively to the past year.

$($b$)$ Use the forecasting method with the smallest MAD value to forecast the average number of these passengers flying in January of the new year.

$17$