When we find confidence intervals for samples means we use the student-t distribution.

Two conditions must be met: the sample must be random and it the sample size must be large enough for the central limit theorem. (Around 30)

The general formula is:

upper limit: sample mean estimate + t-value*standard error

lower limit: sample mean estimate - t-value*standard error

In math this can be

upper limit: X + t*(s/sqrt(n))

lower limit: X - t*(s/sqrt(n))

The t value is determined by the point on the x-axis that represents the amount of probability we want and is found much like the z-value for proportion confidence intervals.

The t-value must also generally be found by using a computer or a table. The t-distribution takes the number of the sample size into account, and calls this accounting "degrees of freedom". Degrees of freedom are n-1, or one less than your sample size. Generally the more degrees of freedom, the better your estimates.

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