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The confidence interval for the expectation of a normal sample

Lecture



The case of known variance [edit]

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

Statement. Random value

  The confidence interval for the expectation of a normal sample

has a standard normal distribution   The confidence interval for the expectation of a normal sample . Let be   The confidence interval for the expectation of a normal sample -   The confidence interval for the expectation of a normal sample -quantile standard normal distribution. Then, due to the symmetry of the latter, we have:

  The confidence interval for the expectation of a normal sample .

After substitution of the expression for   The confidence interval for the expectation of a normal sample and simple algebraic transformations we get:

  The confidence interval for the expectation of a normal sample .

The case of unknown variance [edit]

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

Statement. Random value

  The confidence interval for the expectation of a normal sample ,

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

  The confidence interval for the expectation of a normal sample .

After substitution of the expression for   The confidence interval for the expectation of a normal sample and simple algebraic transformations we get:

  The confidence interval for the expectation of a normal sample .

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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis