### Math 225 Course Notes

### Section 7.5:
Hypothesis Testing: A Population Proportion

A hypothesis about a population proportion takes the form
H_{0}: p = p_{0}

An alternative hypothesis can be one-sided.
H_{A}: p < p_{0}

or
H_{A}: p > p_{0}

Otherwise, an alternative hypothesis can be two-sided
H_{A}: p does not equal p_{0}

In any case, the hypothesis is tested by considering the sample proportion
and its sampling distribution.
For large samples,
the shape of the sampling distribution is approximately normal.
A rule of thumb is that if both np and n(1-p) are larger than 5,
the sample is sufficiently large for the normal approximation.
The result can be compared to the standard normal distribution.
This depends on the central limit theorem.

Suppose that 1000 American women
aged 50--54 are randomly selected,
and that 18 are found to have breast cancer.
Is there evidence that the proportion
of American women in this age groups with breast cancer
is less than 2%?
Since np = 1000(.02) = 20 > 5,
we may use the normal approximation.

We state hypotheses as

H_{0}: p = .02
H_{A}: p < .02

Our sample proportion is 18/1000 = .018.
If the null hypothesis is true,
the SE is
sqrt( (.02)(.98)/1000 ) = .00443

The z statistic for the z-test is
(.018 - .020) / .0043 = -4.65

The p-value is essentially 0.
This is very strong evidence that the proportion of women is below 2%.
However, it is only below 2% by a small amount.
The large sample size allows us to conclude with high confidence
that the population proportion is in a small interval close to 1.8%.

Last modified: April 15, 1996

Bret Larget,
larget@mathcs.duq.edu