### Math 225 Course Notes

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### Chapter 6

#### Contents

In Chapter 5,
we studied the distribution of statistics
(such as the sample mean)
when we knew the population from which the sample was drawn.
In Chapter 6,
we reverse the problem and study
what we can say about a population parameter
on the basis of data from a random sample.
It is never sufficient to simply estimate a parameter with a number.
We should also include a description
of how much error we think there is in our estimation procedure.
We do this with confidence intervals.
The logic for constructing confidence intervals will be the same
for all the examples we consider this semester.
The logic is illustrated in the context of estimating a population mean
with a 95% confidence interval.
- The sampling distribution for
is approximately normally distributed with a mean of
and a standard deviation of
.
- For 95% of all possible samples,
is within
1.96
of .
- Thus, for 95% of all possible samples,
is within
1.96 of
.
- Therefore, I am 95% confident that
is within
1.96
of
for the particular sample I have chosen.

Different contexts will call for using different formulas,
but the basic structure of all confidence intervals we study
this semester will be the same.
(estimate) +/- (reliability coefficient)(standard error)

We will consider confidence intervals
in the four situations from Chapter 5:
the population mean from a single population,
the difference between two population means from
two independent samples,
the population proportion from a single population,
and the difference between two population proportions
from two independent samples.
For estimating population means when population standard deviations
are unknown,
the t distribution should be used
to determine reliability coefficients.

Last modified: Feb 26, 1996

Bret Larget,
larget@mathcs.duq.edu