Consider the following gasoline sales time series data. Click on the datafile logo to reference the data.

Week Sales $(1000$s of gallons$)$

$118$

$221$

$319$

$424$

$518$

$615$

$721$

$819$

$922$

$1019$

$1116$

$1223$

a$.$ Using a weight of for the most recent observation, for the second most recent observation, and for third most recent observation, compute a three$-$week weighted moving average for the time series $($to $2$ decimals$).$ Enter negative values as negative numbers.

Weighted Moving

Week Time$-$Series Value

$(1000$s of gallons$)$ Average Forecast

$(1000$s of gallons$)$ Forecast Error

$(1000$s of gallons$)$

$($Error$)2$

Total

b$.$ Compute the MSE for the weighted moving average in part $($a$)($to $2$ decimals$).$

MSE $=$

Do you prefer this weighted moving average to the unweighted moving average? Remember that the MSE for the unweighted moving average is $.$

Prefer the moving average here; it has a smaller MSE.

c$.$ Suppose you are allowed to choose any weights as long as they sum to $.$ Could you always find a set of weights that would make the MSE at least as small for a weighted moving average than for an unweighted moving average?