Question

How can I use confidence intervals for the population mean µ?

Answer 1

`m+-ts`

Where `t` is the `t`-score associated with the confidence interval you require.
[ If your sample size is greater than 30 then the limits are given by
`mu` = `bar x +-(z xx SE)`]

Explanation:

Calculate the sample mean (`m`) and sample population (`s`) using the standard formulas.

`m=1/Nsum(x_n)`

`s=sqrt(1/(N-1)sum(x_n-m)^2`

If you assume a normally distributed population of i.i.d. (independent identically distributed variables with finite variance) with sufficient number for the central limit theorem to apply (say `N>35`) then this mean will be distributed as a `t`-distribution with `df=N-1`.

The confidence interval is then:

`m+-ts`

Where `t` is the `t`-score associated with the confidence interval you require.

If you know the population standard deviation and do not need to estimate it (`sigma`), then replace `s` with `sigma` and use a Z score from the normal distribution rather than a `t`-score since your estimate will be normally distributed rather than `t` distributed (using the above assumptions about the data).

[`barx` = sample Mean
z = critical value
SE is standard Error
SE = `sigma / sqrt(n)` Where n is sample size.

Upper limit of the population --`mu` = `bar x +(z xx SE)`
Lower limit of the population - `mu` = `bar x -(z xx SE)`

If your sample size is less than 30 use the 't' value]