The confidence interval for the expectation of a normal sample

The case of known variance [edit]

Let be - independent sampling from the normal distribution, where - known dispersion. Define arbitrary and build a confidence interval for the unknown mean .

Statement. Random value

has a standard normal distribution . Let be - -quantile standard normal distribution. Then, due to the symmetry of the latter, we have:

.

After substitution of the expression for and simple algebraic transformations we get:

.

The case of unknown variance [edit]

Let be - independent sampling from the normal distribution, where - unknown constants. Construct a confidence interval for an unknown mean .

Statement. Random value

,

Where - unbiased sample standard deviation, has a Student’s distribution with degrees of freedom . Let be - quantile student distribution. Then, due to the symmetry of the latter, we have:

.

After substitution of the expression for and simple algebraic transformations we get:

.