Tests for the difference between two population proportions

(Large samples)

Next we will develop procedures for comparing two population proportions. We will consider standard model with a random sample of observations with proportion ”successes” and an independent random sample of

observations from population with proportion ”successes”.

We know that for large samples, proportions can be approximated as normally distributed random variables and as a result

has a standard normal distribution.

We want to test the hypothesis that the population proportions and are equal. Denote their common value by , then the value under this hypothesis

follows a good approximation a standard normal distribution.

Finally, unknown common proportion can be estimated by a pooled estimator defined as

or

where and are number of “successes” in and , respectively.

Which of these formulas is used to calculate depends on whether the values of and or the values of and are known.

Testing equality of two population proportions:

We are given independent samples of size and with proportion of successes and . When we assume that the population proportions are equal, an estimate of the common proportion is

or .

For large sample sizes the value of the test statistic for is computed as

Then 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

Example:

A company is planning to buy a few machines. Company is considering two

types of machines, but will buy all of the same type. The company selects

one machine from each type and uses for a few days. A sample of 900 items produced on machine A showed that 55 of them were defective. A sample of 700 items produced on machine B showed that 41 of them were defective. Testing at 1% significance level, can we conclude based on the information from these samples that the proportions of the defective items produced on the two machines are different?

Solution:

Let be the proportion of all items in all items produced on machine A, and be the proportion of all items in all items produced on machine B. Let and be the corresponding sample proportions. Let and be the number of defective items in two samples respectively.

Machine A: ;

Machine B: ;

The two sample proportions are calculated as follows:

;

The null and alternative hypotheses are

(the two proportions are equal)

(the two proportions are different)

The decision rule is

Reject if or

Let us check if the sample sizes are large:

Since the samples are large and independent we apply the normal distribution to make a test.

The pooled sample proportion is

The value of the test statistics is

.

Let us find the value of .

;

and

The value of the test statistic falls in the nonrejection region. Consequently, we fail to reject the null hypothesis. As a result, we can conclude that proportions of defective items produced by two machines are not different.

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