Year Value Forecast "Forecast

Error" "Squared

Forecast

Error" $5.$ Use the time series data to the left in answering questions a$-$d below. $(12$ points$)$

$1234000$

$2287140\mathrm{.}8-146\mathrm{.}221374\mathrm{.}44($a$)$ In the space provided to the left, use $\backslash$alpha $=0\mathrm{.}4$ to compute the exponential smoothing values for the time series. $(5$ pts$)$

$325531\mathrm{.}8-223\mathrm{.}249818\mathrm{.}24$

$4310121\mathrm{.}6-188\mathrm{.}435494\mathrm{.}56$

$529864\mathrm{.}8-233\mathrm{.}254382\mathrm{.}24($b$)$ Compute MSE $=37193\mathrm{.}63556$

$6255114\mathrm{.}4-140\mathrm{.}619768\mathrm{.}36$

$730239-26369169($c$)$ Compute a forecast for year $11=84\mathrm{.}6(2$ pts$)$

$8267142\mathrm{.}6-124\mathrm{.}415475\mathrm{.}36$

$922518-20742849($d$)$ Which method provides a better forecast, two$-$year moving average from question $4$ or exponential smoothing from question $5$ and why? $(3$ pts$)$

$10336117\mathrm{.}4-218\mathrm{.}647785\mathrm{.}96$

total $334742\mathrm{.}72$ Exponential smoothing provides a better forecast because it has a lower MSE, indicating that it produces forecasts that are closer to the actual values compared to the two$-$year moving average. Exponential smoothing gives more weight to recent data, which might capture trends better than a simple moving average.