# Symmetric Confidence Intervals

Investigating Symmetric Confidence Intervals for the mean of a sample from a large non-Normal population, where the population variance is known.

###### Author and programmer: Ron Barrow

UK Year 13, KS5, GCE A Level Mathematics - Statistics

TweetInstructions below See also: The Central Limit Theorem

## How to Use this Applet

Before trying to understand confidence intervals, make sure you understand the Central Limit Theorem. If you do not, then visit the Central Limit Theorem applet on this site.

From the Central Limit Theorem we know that for a large population (in this
case 1500), the distribution of the means of random samples of the same size (in
the starting case 50) is:

1 approximately Normal

2 mean = population mean

3 variance = population variance ÷ sample size
This distribution is shown in yellow on the screen.

When you take a single random sample from the population and calculate its
mean, how close to the population mean is it? How **confident** are
you that they are close to one another? Does it matter how large the sample is?

When you click on "Take a Sample" above a new sample is taken. Sometimes the
mean is close to the population mean, sometimes not. But we can construct
**confidence intervals** from probabilities based on the fact that
the distribution of the sample mean is Normal.

Look at the 50% confidence interval (CI). About half the time the mean of the
population will be in that interval, and about half the time not. The columns on
the right count a 'hit rate', and after a few samples you should see that the
hit rate is running around 50%. Try it and see.

Do the same for the 68%, 90%, 95% and 99% CIs. Vary the sample sample size and see what happens. If you click on "Reset Hits" it sets the hit counts to 0 and takes the first of a new
batch of samples. Try it out.

Start investigating!