Confidence Intervals

Date: 07/24/2006 at 02:12:16
From: Diako
Subject: confidence interval

Dear Dr Math,

According to the correct definition of confidence interval, if n 
experiments were to be repeated many times, 95% of the calculated 
confidence intervals +/-(ts/sqrt(n)) would include the true value. 

In the real world (say in a chemical lab) only n experiments are 
performed and only one of those many confidence intervals is 
calculated and reported.  What does this single CI mean?  How can this 
single range represent the population?  Customers usually misinterpret
this as a range in which the true value lies with 95% probability!

Date: 07/26/2006 at 02:27:06
From: Doctor Wilko
Subject: Re: confidence interval

Hi Diako,

Thanks for writing to Dr. Math!

It's confusing and the distinction seems subtle, but I'll try to
explain my understanding of this.

Why do we need confidence intervals?
===================================

If you calculate a sample mean, you have a point estimate.  Is this a
good estimate of the true population mean?  A bad estimate?  Who 
knows?  

The point estimate is easy to calculate and interpret, but we don't
know how accurate it is.  Instead it might be more meaningful to look
at an interval estimate, i.e., the confidence interval.

So, what is a confidence interval?
=====================================

It's a RANGE of values used to estimate the true value of a population
parameter.

The degree of confidence (e.g., 90%, 95%, etc...) tells us the
percentage of times that the confidence interval actually contains the
population parameter, assuming the process is repeated a large number
of times.

Incorrect Interpretation:
========================

The confusion with interpreting confidence intervals is that people
often draw a sample, calculate the mean, construct a single confidence
interval and say, "There is a 90% probability that the true mean is
within THIS confidence interval."  

This is wrong because the population parameter, mu, is a constant, not
a random variable. Its either in the confidence interval or it's not.
There is no probability involved. 

Correct Interpretation:
======================

Confidence intervals are best interpreted in the context of many
samples.  

Say you construct one confidence interval.  Your boss asks you to
interpret the answer.  You say, "I'm 90% confident that the interval
from ___ to ___ actually contains the true population mean."  Your
boss says, "I'm a little confused, please explain."  You say, "If I
take 100 random samples each of size n and construct the confidence
intervals, then about 90 of them will contain the true population mean."  

In summary, it really alludes to the process.  A confidence level of
90% tells us the process we are using will, in the long run, result in
confidence interval limits that contain the population parameter 90%
of the time.

Does this help?

- Doctor Wilko, The Math Forum
  http://mathforum.org/dr.math/