Statistics

# 10.3Comparing Two Independent Population Proportions

Statistics10.3 Comparing Two Independent Population Proportions

When conducting a hypothesis test that compares two independent population proportions, the following characteristics should be present:

1. The two independent samples are simple random samples that are independent.
2. The number of successes is at least five, and the number of failures is at least five, for each of the samples.
3. Growing literature states that the population must be at least 10 or 20 times the size of the sample. This keeps each population from being over-sampled and causing incorrect results.

Comparing two proportions, like comparing two means, is common. If two estimated proportions are different, it may be due to a difference in the populations or it may be due to chance. A hypothesis test can help determine if a difference in the estimated proportions reflects a difference in the population proportions.

The difference of two proportions follows an approximate normal distribution. Generally, the null hypothesis states that the two proportions are the same. That is, H0: pA = pB. To conduct the test, we use a pooled proportion, pc.

The pooled proportion is calculated as follows:

$p c = x A + x B n A + n B . p c = x A + x B n A + n B .$

The distribution for the differences is

$P ′ A − P ′ B ~N[0, p c (1− p c )( 1 n A + 1 n B ) ]. P ′ A − P ′ B ~N[0, p c (1− p c )( 1 n A + 1 n B ) ].$

The test statistic (z-score) is

$z= ( p ′ A − p ′ B )−( p A − p B ) p c (1− p c )( 1 n A + 1 n B ) . z= ( p ′ A − p ′ B )−( p A − p B ) p c (1− p c )( 1 n A + 1 n B ) .$

### Example 10.8

Two types of medication for hives are being tested to determine if there is a difference in the proportions of adult patient reactions. Twenty out of a random sample of 200 adults given Medication A still had hives 30 minutes after taking the medication. Twelve out of another random sample of 200 adults given Medication B still had hives 30 minutes after taking the medication. Test at a 1 percent level of significance.

Try It 10.8

Two types of valves are being tested to determine if there is a difference in pressure tolerances. Fifteen out of a random sample of 100 of Valve A cracked under 4,500 psi. Six out of a random sample of 100 of Valve B cracked under 4,500 psi. Test at a 5 percent level of significance.

### Example 10.9

A research study was conducted about gender differences in texting. The researcher believed that the proportion of girls involved in texting is less than the proportion of boys involved. The data collected in spring 2010 among a random sample of middle and high school students in a large school district in the southern United States is summarized in Table 10.10. Is the proportion of girls sending texts less than the proportion of boys texting? Test at a 1 percent level of significance.

MalesFemales
Sent texts 183 156
Total number surveyed 2231 2169
Table 10.10

### Example 10.10

Researchers conducted a study of smartphone use (Phone A versus Phone B) among adults. A cell phone company claimed that Phone B smartphones are more popular with whites (non-Hispanic) than with African Americans. The results of the survey indicate that of the 232 African American cell phone owners randomly sampled, 5 percent own Phone B. Of the 1,343 white cell phone owners randomly sampled, 10 percent own Phone B. Test at the 5 percent level of significance. Is the proportion of white Phone B owners greater than the proportion of African American Phone B owners?

Try It 10.10

A group of citizens wanted to know if the proportion of homeowners in their small city was different in 2011 than in 2010. Their research showed that of the 113,231 available homes in their city in 2010, 7,622 of them were owned by the families who live there. In 2011, 7,439 of the 104,873 of the available homes were owned by city residents. Test at a 5 percent significance level. Answer the following questions:

a. Is this a test of two means or two proportions?

b. Which distribution do you use to perform the test?

c. What is the random variable?

d. What are the null and alternative hypotheses? Write the null and alternative hypotheses in symbols.

e. Is this test right-, left-, or two-tailed?

f. What is the p-value?

g. Do you reject or not reject the null hypothesis?

h. At the ______ level of significance, from the sample data, there ______ (is/is not) sufficient evidence to conclude that ______.