Consider the model

$(1-0\mathrm{.}43B)(1-B)Z_t=a_t,$

and the observations $Z_{49}=33\mathrm{.}4$ and $Z_{50}=33\mathrm{.}9.$

$($a$)$ Compute the forecast $Z_{50}(l),$ for $l=1,2,3,4,$ and their $90\%$ forecast

intervals.

$($b$)$ At time $t=51,Z_{51}$ became known and equaled $34\mathrm{.}1.$ Updated the

forecast obtained in $($a$).$