Changes

==Example==

==Example==

Define the following:

Define the following mean estimator:

<math>\tilde \mu=\frac{\sum_{i=1}^n X_i}{n}.</math>

<math>\tilde \mu=\frac{\sum_{i=1}^n X_i}{n}.</math>

<math>\tilde \mu \sim G(\mu, \frac{\sigma}{\sqrt n})</math>

<math>\tilde \mu \sim G(\mu, \frac{\sigma}{\sqrt n})</math>

Assume the standard notation (''sigma'' is the [[standard deviation]], ''n'' is the [[sample size]]). Assume we want to find the probability that the difference between the mean and its estimator is less than 100. Then:

Assume the standard notation (''sigma'' is the [[standard deviation]], ''n'' is the [[sample size]]). You are given that sigma/sqrt(n) is 67.5. Assume we want to find the probability that the difference between the mean and its estimator is less than 100. Then:

<math>\Pr(|\tilde \mu - \mu| < 100)=2\Pr(\tilde \mu - \mu > 100)</math>

<math>\Pr(|\tilde \mu - \mu| < 100)=2\Pr(\tilde \mu - \mu > 100)</math>

<math>1 - \Pr(\tilde \mu - \mu < 100)</math>

<math>\Pr(\tilde \mu - \mu > 100)</math><br/>

<math>1 - \Pr(\frac{\tilde \mu - \mu}{\sigma / \sqrt n} > \frac {100}{\sigma / \sqrt n})</math>

<math>=1 - \Pr(\tilde \mu - \mu < 100)</math><br/>

<math>

<math>=1 - \Pr(\frac{\tilde \mu - \mu}{\sigma / \sqrt n} > \frac {100}{\sigma / \sqrt n})</math><br/>

<math>\approx 1 - \phi(\frac{100}{67.5})</math><br/>

<math>= 0.069</math>, where ''Phi'' is the [[cumulative distribution function]] for the G(0,1) Gaussian distribution.