**Inference For Proportions:**

The
proportion of a population having a given characteristic is a parameter, *p*.
The proportion of a sample having a given characteristic is a statistic,
_{}.

We know that the sampling
distribution of _{} is approximately
normal (for sufficiently large samples) with mean _{} and standard deviation
_{}.

To standardize _{}, subtract the mean and divide by the standard
deviation. This gives the z test
statistic:

_{}

For a hypothesis test where _{}, use _{} to estimate _{}.

For a confidence
interval, use _{} as an estimate of _{}.

Assumptions for inference
about a proportion:

1. SRS

2. population ³ 10n

3. _{} ³
10 and _{} ³ 10 for a hypothesis test

_{} ³ 10 and _{} ³ 10 for a confidence interval

The margin of error is: _{}

So to determine the sample
size needed for a specified margin of error, let

_{} and solve for n.

If we want to compare two
populations or compare the responses to two treatments from independent
samples, we look at a two-sample proportion.

The null hypothesis is that
there is no difference between the two parameters.

_{} or _{}

The alternative hypothesis
could be that

_{} (two-sided)

_{} or _{} (one-sided)

_{} or _{} (one-sided)

Before you begin, check your
assumptions!

Assumptions for inference
about a proportion:

1. SRS

2. independently chosen samples

3. population ³ 10n

4. _{}, _{}, _{}, _{} ³ 5 for a hypothesis test

_{}, _{}, _{}, _{} ³ 5 for a C.I.

If these assumptions hold,
then the difference in sample proportions is an unbiased estimator of the
difference in population proportions, so
_{} is equal to _{}.

Also, the variance of _{} is the sum of the
variances of _{} and _{}, which is _{}.

In order to standardize _{}, subtract the mean and divide by the standard error:

_{}

Use this test statistic to
carry out a test of significance.

On the TI-83+, use
2-PropZInt and 2-PropZTest to construct confidence intervals and perform
significance tests.