# Confidence interval of mean

• Jul 26th 2017, 06:50 AM
xl5899
Confidence interval of mean
Both images are 2 consecutive pages of my notes . In this theory , i was told to use z-distribution when the sample size is large ( more than 30) and the standard deviation of the population , σ is known) . However , in the 2nd image , i was told to replace σ with s ( sample standard deviation) . Which is correct ?

I think the first theory of using z-distribution when the sample size is large ( more than 30) and the standard deviation of the population , σ is known) is correct , and the replace σ with s ( sample standard deviation) . is wrong . Am i right ?

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• Aug 1st 2017, 10:45 AM
abender
Re: Confidence interval of mean
Quote:

Originally Posted by xl5899
Both images are 2 consecutive pages of my notes . In this theory , i was told to use z-distribution when the sample size is large ( more than 30) and the standard deviation of the population , σ is known) . However , in the 2nd image , i was told to replace σ with s ( sample standard deviation) . Which is correct ?

I think the first theory of using z-distribution when the sample size is large ( more than 30) and the standard deviation of the population , σ is known) is correct , and the replace σ with s ( sample standard deviation) . is wrong . Am i right ?

Attachment 37906Attachment 37907

It is always better to use $\displaystyle \sigma$ instead of $\displaystyle s$; however, in practice you will often be given a large set of data and will only be able to compute $\displaystyle s$. When your data are large enough, you may approximate $\displaystyle \sigma$ with $\displaystyle s$.

Bottom line: Use $\displaystyle \sigma$ if you have it, but don't hesitate to use $\displaystyle s$ if your data are large as an approximation to $\displaystyle \sigma$.

I hope this clears things up for you.
Best,
Andy
• Aug 1st 2017, 10:48 AM
abender
Re: Confidence interval of mean
Quote:

Originally Posted by abender
It is always better to use $\displaystyle \sigma$ instead of $\displaystyle s$; however, in practice you will often be given a large set of data and will only be able to compute $\displaystyle s$. When your data are large enough, you may approximate $\displaystyle \sigma$ with $\displaystyle s$.

Bottom line: Use $\displaystyle \sigma$ if you have it, but don't hesitate to use $\displaystyle s$ if your data are large as an approximation to $\displaystyle \sigma$.

I hope this clears things up for you.
Best,
Andy

Let me add one more bit to this. The Z-test assumes the standard deviation(s) is(are) known.