# Confidence Intervals: t-distribution or normal distribution?

1. Oct 19, 2010

### Richard_R

Hi all,

When working out confidence intervals based on population samples are you supposed to always use t-distributions, standard normal (z) distributions, or do you make a choice based on the sample size?

Up until now I've been lucky enough to have large sample sizes (for some work I'm doing) so have been using the z-distribution. However I now have some data sets which range from n=1 (lol) to n=29 so am not sure if I should now be using t-distributions to define confidence intervals, or how I'd make that decision (e.g. use t-distribution if n<30, for example?)

Thanks
-Rob

2. Oct 22, 2010

### SW VandeCarr

Assuming the normal assumption is valid, the general rule is to use the t-distribution to calculate confidence intervals where the number of degrees of freedom (df=n-1) is less then 30, The Z and t scores are similar around this value. Skewed data, particularly in small samples, make CIs fairly useless. In larger samples, normalizing transformations can be useful for constructing CIs..

3. Oct 22, 2010

Actually the notion of using the sample size as the determining factor is being (as it should be) tossed out. It is a remnant of the days before computing power was so readily available.

IF the assumption of normality can be made, when you know $$\sigma$$ (population standard deviation) use the Z-interval. When you don't know sigma (so you have only the sample standard deviation) use the t-interval.

If your data is badly skewed, it is debatable whether the mean is the appropriate parameter to measure central tendency.

4. Oct 22, 2010

### SW VandeCarr

Well I am retired and involved in other things, but I have researched the t distribution recently and I've not run across this. However, my research was mostly on the math and not the application.

What you say makes sense. Would you use the Z value for very small samples, say n=5, if you did know sigma?

EDIT: In most of my experience sigma is not known.

Last edited: Oct 22, 2010
5. Oct 22, 2010