Confidence Intervals: t-distribution or normal distribution?


by Richard_R
Tags: confidence, distribution, intervals, normal, tdistribution
Richard_R
Richard_R is offline
#1
Oct19-10, 03:46 AM
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
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SW VandeCarr
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#2
Oct22-10, 08:32 AM
P: 2,490
Quote Quote by Richard_R View Post
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..
statdad
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#3
Oct22-10, 10:42 AM
HW Helper
P: 1,344
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 [tex] \sigma [/tex] (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.

SW VandeCarr
SW VandeCarr is offline
#4
Oct22-10, 04:48 PM
P: 2,490

Confidence Intervals: t-distribution or normal distribution?


Quote Quote by statdad View Post
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 [tex] \sigma [/tex] (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.
statdad
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#5
Oct22-10, 10:40 PM
HW Helper
P: 1,344
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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