Question

Suppose that you want to perform a hypothesis test for a population mean based on a small sample but that preliminary data analyses indicate either the presence of outliers or that the variable under consideration is far from normally distributed.

a. Is either the z-test or t-test appropriate ?

b. If not, what type of procedure might be appropriate ?

Short answer

Part (a) Neither the z-test procedure not the t-test procedure is appropriate.

Part (b) If either t-test procedure or z-test procedure is not appropriate, then use the non-parametric test.

Step-by-step solution

 Part (a) Step 1. Given information

The given tests aret-test andz-test

Part (a) Step 2. Is either the z-test or t- test appropriate ? 

Conditions to use of thez-test procedure are given below:
Small Sample size:
If the sample size is less than 15, thet-test procedure is used when the variable is normally distributed or very close to being normally distributed.
Moderate Sample size:
If the sample size lies between 15 and 30 , thet-test procedure is used when the variable far from being normally distributed or there is no outlier in the data.
Large Sample size:
If the sample size is greater than 30 , the t-test procedure is used without any restriction.

Conditions to use of thet-test procedure are given below:
Small Sample size:

• Samples are randomly selected from the population.
• Population follows normal distribution or the sample size is larger.
• The standard deviation is unknown.

Explanation:
Here, the sample is selected from the population and the sample size is small. Moreover, there is either the presence of outliers. Also, the distribution of the variable is far from normal distribution. Hence, from the above conditions, it is clear that neither the z-test procedure not thet-test procedure is appropriate.

Part (b) Step 1. Given information

The given tests aret-test andz-test

Part (b) Step 2. If not, what type of procedure might be appropriate ? 

If eitherz-test procedure ort-test procedure is not appropriate, then use the non-parametric test.
Reason for procedure is not appropriate:
If the data is affected by the outliers or the non-normality, then only the non-parametric test is appropriate.