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Chart and Diagram for Confidence Interval Example

This chart and diagram may help clarify the terminology and ideas involved in understanding the concept of confidence interval.

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
y1, y2, ... , yn
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 = (y1+ y2+ ... + yn)/n 1) Each sample has its own mean ȳ. This allows us to define a random variable Ȳn. The population for Ȳn is all simple random samples from Y. The value of Ȳn for a particular simple random sample is the sample mean ȳ for that sample.

2)  Since it is a random variable,  
 Ȳn also has a mean, E( Ȳn). Using the model assumptions  for this particular example, it can be proved that E( Ȳn) = µ. In other words, Y and Ȳn have the same mean as random variables.
Associated Distribution Distribution of Y None Sampling Distribution (Distribution of Ȳn )

The diagram shows the distribution of Y (in blue) and the sampling distribution (distribution of  Ȳn, in red) when Y is standard normal and n = 9. Notice that both distributions have the same mean.

Distribution of Y and sampling distribution of mean for samples of size n = 49 from Y
Last updated May 7, 2012