Part $2$:

Mr Sweet provided Daniella with some data on his bakery shop. MyCakes also produces chocolate cakes, and this data is recorded week by week. Mr Sweet said: $$In the last year we worked hard on a promotional campaign to retain loyal customers by offering them vouchers with discounts. This both increased the sales and evened them out: check our numbers!$$ Daniella is not sure how to do that, so she asks you to review the data on her behalf.

Using the data below Choco cake sales, calculate the weekly average chocolate cake sales for years $2022$ and $2023?$

Year week number of cakes sold

$20221111$

$20222137$

$20223143$

$20224124$

$2022581$

$2022656$

$20227105$

$2022882$

$20229105$

$20221096$

$20221195$

$20221275$

$20221394$

$202214119$

$20221574$

$202216116$

$202217115$

$20221896$

$202219118$

$20222064$

$20222195$

$202222122$

$202223108$

$202224102$

$20222584$

$20222669$

$20222799$

$20222880$

$20222976$

$20223059$

$20223163$

$20223271$

$20223361$

$20223471$

$202235112$

$20223677$

$202237118$

$20223897$

$202239106$

$202240107$

$202241123$

$20224279$

$20224375$

$20224499$

$202245125$

$20224695$

$202247146$

$202248103$

$20224972$

$20225098$

$202251131$

$20225279$

$20231115$

$20232147$

$20233155$

$20234145$

$2023598$

$2023671$

$20237106$

$2023885$

$20239107$

$20231096$

$20231195$

$20231289$

$20231397$

$202314129$

$20231595$

$202316115$

$202317134$

$202318112$

$202319131$

$20232072$

$202321102$

$202322121$

$202323110$

$202324114$

$202325106$

$20232673$

$202327117$

$20232892$

$20232977$

$20233083$

$20233187$

$20233295$

$20233371$

$20233482$

$202335125$

$20233681$

$202337140$

$202338121$

$202339126$

$202340116$

$202341126$

$20234289$

$20234392$

$202344121$

$202345141$

$202346112$

$202347150$

$202348122$

$20234985$

$202350101$

$202351135$

$20235291$

For each year, calculate the $95\%$ and $90\%$ confidence intervals of the weekly average sales?

Discuss the results of your analysis with respect to the level and the variability of weekly sales in $2022$ and $2023.$ Discuss the factors that influence the width of the confidence intervals.