# Confidence interval of mean

1. Jul 26, 2017

### tzx9633

1. The problem statement, all variables and given/known data
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 ?

2. Relevant equations

3. The attempt at a solution
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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2. Jul 26, 2017

### FactChecker

If you know the value of σ from some theory or some larger external set of data, then use it. Otherwise, you have to use your sample data to estimate σ. That tends to give a slightly smaller estimate than it should for the general population, so a better estimate of σ is obtained by dividing by n-1 instead of n.

3. Jul 26, 2017

### Ray Vickson

I would say the second attachment is wrong: the correct distribution to use is the t-distribution with $n-1$ degrees of freedom. However, for large $n$ there is not much difference between the t-distribution and the normal distribution, so it is a reasonable approximation to replace the t by the normal. For example, when $n=31$ (30 degrees of freedom) we have
$$\begin{array}{cc} P(N \leq -1) = 0.15866, & P(T \leq -1) = 0.16265\\ P(N \leq 1) = 0.84134, & P(T \leq 1) = 0.83735 \end{array}$$

4. Jul 28, 2017

### tzx9633

Do you mean that for large sample size n > 30 , it's okay to use the assumption that the standard deviation of the population equal to the standard deviation of the sample ?

5. Jul 28, 2017

### tzx9633

Do you mean that for large sample size n > 30 , it's okay to use the assumption that the standard deviation of the population equal to the standard deviation of the sample ?

6. Jul 28, 2017

### FactChecker

For any size sample, if you don't have some other way of knowing the value of σ, then you have no alternative to using the sample standard deviation.