# Confidence Intervals: t-distribution or normal distribution?

by Richard_R
Tags: confidence, distribution, intervals, normal, tdistribution
 P: 14 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
P: 2,490
 Quote by 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? Thanks -Rob
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..
 HW Helper P: 1,334 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.
P: 2,490

## Confidence Intervals: t-distribution or normal distribution?

 Quote by statdad 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.
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.
 HW Helper P: 1,334 If the sample size is only 5 i would be hesitant to do any confidence interval but, if pushed, if sigma were known, and if told that the data were known to be normally distributed, the Z-interval would be appropriate.

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