CK-12 Probability and Statistics - Advanced

http://www.ck12.org Chapter 8. Hypothesis Testing

Testing a Proportion Hypothesis Using the P-Value

Similar to testing hypotheses dealing with population means, we use a similar set of steps when testing proportion
hypotheses.

1. Determine and state the null and alternative hypotheses.


2. Set the criterion for rejecting the null hypothesis.


3. Calculate the test statistic.


4. Interpret the results and decide whether to reject or fail to reject the null hypothesis.

To test a proportion hypothesis, we use the formula for calculating the test statistic for a mean, but modify it
accordingly. Therefore, our formula for the test statistic of a proportion hypothesis is:

z=

p−P

sp

where:

p=the sample proportion

P=the hypothesized population proportion

sp=the standard error of the proportion

Example:

A congressman is trying to decide on whether to vote for a bill that would legalize gay marriage. He will decide
to vote for the bill only if 70 percent of his constituents favor the bill. In a survey of 300 randomly selected voters,
224 ( 74 .6%)indicated that they would favor the bill. Should he vote for the bill or not?

Solution:

First, we develop our null and alternative hypotheses.

H 0 :P= 0. 70

Ha:P> 0. 70

Next, we should set the criterion for rejecting the null hypothesis. We will use a probability (?) level of 0.05 and
since we are interested only in the probability that the percentage of constituents isgreaterthan 0.70, we will use a
single-tailed test. Looking at the standardz-table, we find that thecritical valuefor a single-tailed test at an alpha
level of 0.05 is equal to 1. 64.

To calculate the test statistic, we first find the standard error of the proportion.

Sp=

√

PQ

n

=

√

0. 70 × 0. 30

300

≈ 0. 0265

After finding the standard error, we can calculate the standardz-score needed to evaluate our hypothesis.

z=

p−P

sp

=

0. 74 − 0. 70

0. 0265

≈ 1. 51