To solve this problem, I will:

Compute the three$-$period and four$-$period moving averages for weeks $5,6,$ and $7.$

Calculate the mean absolute deviation $($MAD$)$ for each forecasting method to determine which is more accurate.

Use the more accurate method to forecast week $8.$

Three$-$Period Moving Average:

Week $5$ forecast $=(27+31+19)/3=25\mathrm{.}67$

Week $6$ forecast $=(31+19+17)/3=22\mathrm{.}33$

Week $7$ forecast $=(19+17+21)/3=19$

Four$-$Period Moving Average:

Week $5$ forecast $=(21+27+31+19)/4=24\mathrm{.}5$

Week $6$ forecast $=(27+31+19+17)/4=23\mathrm{.}5$

Week $7$ forecast $=(31+19+17+21)/4=22$

Calculating MAD for Three$-$Period:

$|$Forecast Error$|=|$Actual $-$ Forecast$|$

Week $5$: $|19-25\mathrm{.}67|=6\mathrm{.}67$

Week $6$: $|17-22\mathrm{.}33|=5\mathrm{.}33$

Week $7$: $|21-19|=2$

MAD $=(6\mathrm{.}67+5\mathrm{.}33+2)/3=4\mathrm{.}67$

Calculating MAD for Four$-$Period:

Week $5$: $|19-24\mathrm{.}5|=5\mathrm{.}5$

Week $6$: $|17-23\mathrm{.}5|=6\mathrm{.}5$

Week $7$: $|21-22|=1$

MAD $=(5\mathrm{.}5+6\mathrm{.}5+1)/3=4\mathrm{.}33$

The four$-$period moving average has a lower MAD of $4\mathrm{.}33,$ so it is the more accurate forecasting method.

To forecast week $8$ using the four$-$period method:

Week $8$ forecast $=(19+17+21+$ X$)/4$

Where X is the actual sales for week $7,$ which we don't know.

So using just the known sales data:

Week $8$ forecast $=(19+17+21)/3=19$

Therefore, using the more accurate four$-$period moving average method, the forecast for week $8$ sales is $19$ cases.