Confidence Intervals: t-distribution or normal distribution?

In summary, when working with confidence intervals based on population samples, you should use the t-distribution if the number of degrees of freedom (df=n-1) is less then 30. If the assumption of normality can be made, use the Z-interval. If the data is badly skewed, it is debatable whether the mean is the appropriate parameter to measure central tendency.
  • #1
Richard_R
14
0
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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  • #2
Richard_R said:
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..
 
  • #3
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.
 
  • #4
statdad said:
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.
 
Last edited:
  • #5
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.
 

1. What is a confidence interval and why is it important in statistical analysis?

A confidence interval is a range of values that is likely to contain the true value of a population parameter with a certain level of confidence. It is important in statistical analysis because it helps to quantify the uncertainty in our estimation of a population parameter and allows us to make more accurate conclusions about the population.

2. What is the difference between using a t-distribution and a normal distribution for calculating confidence intervals?

The t-distribution is used when the population standard deviation is unknown and must be estimated from the sample. The normal distribution is used when the population standard deviation is known. The t-distribution has heavier tails compared to the normal distribution, making it more suitable for smaller sample sizes.

3. How is the confidence level chosen for a confidence interval?

The confidence level is chosen based on the level of uncertainty that is acceptable in the estimation of the population parameter. A common confidence level is 95%, which means there is a 95% probability that the true population parameter falls within the calculated confidence interval.

4. Can a confidence interval be used to make a prediction about an individual in the population?

No, a confidence interval is used to make a prediction about the population parameter, not an individual. It is not possible to determine with certainty whether an individual falls within the confidence interval.

5. How does sample size affect the width of a confidence interval?

The larger the sample size, the narrower the confidence interval. This is because a larger sample size provides more precise estimates of the population parameter, resulting in a smaller margin of error and a narrower confidence interval.

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