We have been provided a scenario for a hypothesis test for a population mean. Decide whether the z-test is an appropriate method for conducting the hypothesis test. Assume that the population standard deviation is known in the given case.

Preliminary data analyses reveal that the sample data contain an outliner. It is determined that the outlier is a legitimate observation and should not be removed. The sample size is$17$.

First, we need to make sure that the z-test procedure is suitable for the given sample size. To use the z-test, we need to take care of the following conditions,

In case the sample size is$\left(n\right)<15$, then the z-test procedure can be used when the variable is very close to being normally distributed or normally distributed.

If the sample size is between $15<\left(n\right)<30$, then the z-test procedure can be used when there is no outlier in the data or the variable is far from being normally distributed.

If the sample size is greater than $30<\left(n\right)$, then the z-test procedure can be used without any limitation.

Here, the given sample size is small. This means, the sample size $\left(n\right)$is $17\left(<30\right)$. As there is outliers appearing in the data.

Hence, the use of the z-test is inappropriate in this scenario.