Confidence Interval for μ, Standard Deviation Estimated (1 of 3)

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It is very rare for a researcher wishing to estimate the mean of a population to already know its standard deviation. Therefore, the construction of a confidence interval almost always involves the estimation of both μ and σ.

When σ is known, the formula:

M - zσM ≤ μ ≤ M + zσM

 is used for a confidence interval. When σ is not known,

s divided by the square root of N (N is the sample size)

 is used as an estimate of σM. Whenever the standard deviation is estimated, the t rather than the normal (z) distribution should be used. The values of t are larger than the values of z so confidence intervals when σ is estimated are wider than confidence intervals when σ is known

The formula for a confidence interval for μ when σ is estimated is:

M - t sM ≤ μ ≤ M + t sM

where M is the sample mean, sM is an estimate of σM, and t depends on the degrees of freedom and the level of confidence.
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