The example: Y is a normal random variable; we are interested in a confidence interval for the mean µ of Y.

Population | One Simple Random
Sample y _{1},
y_{2}, ... , y_{n} |
All Simple Random Samples of size n | |

Associated Mean(s) | Population mean µ, also called E(Y), or the expected value of Y, or the expectation of Y. | Sample mean ȳ = (y_{1}+
y_{2}+ ... + y_{n})/n |
1) Each sample has its own mean ȳ.
This allows us to define a random variable Ȳ_{n}.
The population for Ȳ_{n}
is all simple random samples from Y. The
value of Ȳ_{n }for
a particular simple random sample is the sample mean ȳ
for that sample.2) Since it is a random variable, Ȳ _{n
}also has a mean, E( Ȳ_{n}).
Using the model assumptions for this particular example, it can
be proved that E( Ȳ_{n})
= µ. In other words, Y and Ȳ_{n}
have the same mean as random variables. |

Associated Distribution | Distribution of Y | None | Sampling Distribution (Distribution of Ȳ_{n}
) |

The diagram shows the distribution of Y (in blue) and the sampling distribution (distribution of Ȳ

Last updated May 7, 2012