Use the case below to answer the following question$($s$)$

A certain restaurant located in a resort community is owned and operated by Karen Payne. The restaurant just completed its third year of operation. During this time, Karen sought to establish a reputation for the restaurant as a high quality$-$dining establishment that specialises in fresh seafood. The efforts made by Karen and her staff proved successful, and her restaurant is currently one of the best and fastest$-$growing restaurant in their neighbourhood.

Karen concluded that, to plan better for the growth of the restaurant in the future, she needs to develop a system that will enable her to forecast food and beverage sales by month for up to one year in advance. Karen compiled the following data on total food and beverages sales for the three years of operation:

Table $2\mathrm{.}1$: Food and beverage sales for the restaurant $($R$1000$s$)$

Month

First year

Second year

January

$282$

$263$

February

$255$

$238$

March

$265$

$247$

April

$205$

$193$

May

$210$

$193$

June

$160$

$149$

July

$166$

$157$

August

$174$

$161$

September

$126$

$122$

October

$148$

$130$

November

$173$

$167$

December

$235$

$230$

Perform an analysis of the sales data for the restaurant by answering the following questions.

$($a$)$

Provide a graph of the time series sales data $(5$ marks$)$

$($b$)$

Determine a forecasting model to predict the third$-$year sales figures by using linear regression. $(5$ marks$)($c$)$

Stating any assumptions you have made, provide forecast sales for January through December of the third year. $(5$ marks$)$

$($d$)$

If the actuals observed for the third year were as given in Table $2\mathrm{.}2,$ analyse the forecasting errors by answering the following sub$-$questions:

Table $2\mathrm{.}2$: Observed food and beverage sales $($R$1000$s$)$

Month

Third year

January

$242$

February

$235$

March

$232$

April

$178$

May

$184$

June

$140$

July

$145$

August

$152$

September

$110$

October

$130$

November

$152$

December

$206$

i$.$

Determining the forecast error for each month. $(5$ marks$)$

ii$.$

Calculate the mean absolute deviation $($MAD$)$ as a measure of forecasting accuracy. Hence comment on the suitability of your forecasting approach. $(5$ marks$)$ iii.

Provide a graph of the errors, and comment on their behaviour.