Tests of the population proportion (Large sample)

Often we want to conduct test of hypothesis about a population proportion.

This section presents the procedure to perform tests of hypothesis about the population proportion. p for large samples (. The procedure to make such tests is similar in many respects to the one for the population mean .

The value of the test statistic for the sample proportion computed as

where -is the sample proportion, and the value of used in this formula is the one used in the null hypothesis.

Then, if the number of sample observations is large and observed proportion is , the following tests have significance level :

1. To test either null hypothesis

or against the alternative

the decision rule is

Reject if

2. To test either null hypothesis

or against the alternative

the decision rule is

Reject if

3. To test the null hypothesis

against the two sided alternative

the decision rule is

Reject if or

.

Once again, is the number for which

and is the standard normal distribution.

Example:

Mr. A and Mr. B are running for local public office in a large city. Mr. A says that only 30% of the voters are in favor of a certain issue, a law to sell liquor on Sundays. Mr. B doubts A’s statement and believes that more than 30% favor such legislation. Mr. B pays for an independent organization to make a study of this situation. In a random sample 400 voters, 160 favored the legislation. What conclusions should the polling organization report to Mr. B?

Solution:

Let be proportion of all people who favor such legislation and the corresponding sample proportion. Then from given information,

; ; . Let .

The null and alternative hypotheses are as follows

The decision rule is to reject the null hypothesis in favor of alternative if

; .

.

and

From the given information we calculate the value of test statistic as

Since we reject . We make conclusion that more than 30% of voters are in favor of a law to sell liquor on Sundays.

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