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8.4: Comparing Proportions from Two Populations

In Chapter 7 we learned how to use a confidence interval to estimate the difference between two population proportions. Now we will consider how to use a hypothesis test to compare two population proportions.

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Two­Proportion z­test:Suppose p1 and p2 are two different population proportions that we wish to compare. In performing a hypothesis test, the null hypothesis will always be that the proportions are equal:

H0: p1 = p2The alternative hypothesis can be one of three statements:

Ha: p1 ≠ p2Ha: p1 > p2Ha: p1 < p2

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The test statistic we will use is the two­proportion z­test statistic, which is based on the difference in the sample proportions .

The null value is zero, because the null hypothesis always assumes the two proportions are equal and therefore have a difference of zero. If x1 and x2 represent the number of success in the two samples, and n1 and n2 are the sample sizes, then the standard error is given by the formula:

where

The value is called the pooled sample proportion.

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The following conditions let us assume the two­proportion z­test statistic distribution can be approximated by a standard Normal distribution:1. The samples must both be collected randomly and the

observations within the samples must be independent.2. The samples must be independent from each other.3. Both sample sizes are large enough to expect at least 10

successes and 10 failures based on the pooled sample proportion . This means:

, , , and

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Ex 1: Rhino viruses typically cause common colds. In a test of the effectiveness of Echinacea, 40 of the 55 subjects treated with Echinacea developed rhinovirus infections. In a placebo group, 88 of the 103 subjects developed rhinovirus infections. Perform a hypothesis test to determine if Echinacea reduces the rate of rhinovirus infections using a 0.05 significance level.Step 1: Hypothesize

Step 2: Prepare

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Step 3: Compute and compare

Step 4: Interpret

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Ex 2:Use Minitab to perform the hypothesis test in Example 1.Open Minitab. Select Stat, Basic Statistics, 2 Proportions. Choose Summarized data, and enter Number of events and Number of trials for each sample.Click Options…, then make sure Confidence level is set to 95% and Hypothesized difference is 0.0. Select Alternative hypothesis: Difference < hypothesized difference, Test method: Use the pooled estimate of the proportion. Click OK, then OK.

Test and CI for Two Proportions

Sample X N Sample p1 40 55 0.7272732 88 103 0.854369

Difference = p (1) ­ p (2)Estimate for difference: ­0.12709695% upper bound for difference: ­0.0129678Test for difference = 0 (vs < 0): Z = ­1.94 P­Value = 0.026

Fisher’s exact test: P­Value = 0.044

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Ex 3: Researchers wanted to study how people of different ages use cell phones. A random sample of 258 cell phone users ages 20 to 39 were asked if they use their cell phones to stay connected while they are in bed; 168 answered yes. The same question was asked of 129 cell phone users age 40 to 49; 61 answered yes. Is there evidence that younger people use cell phones more in bed than older people? Test the hypothesis at a 0.05 significance level.Step 1: Hypothesize

Step 2: Prepare

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Step 3: Compute and compare

Step 4: Interpret

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Ex 4: Construct a 95% confidence interval for Example 3 to estimate how large of a difference there is.

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Selecting the Appropriate Process:Ex 5: In each case, name the population(s) and then choose whether the appropriate procedure to use is a confidence interval, a one­proportion z­test, or a two­proportion z­test.(a) A researcher takes a random sample of 4­year­olds to find out whether girls or boys are more likely to know the alphabet.

(b) A pollster takes a random sample of U.S. adult voters to see whether more than 50% approve of the performance of the current president.

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(c) A researcher wants to know whether a new heart medicine reduces the rate of heart attacks compared to an old medicine.

(d) Have the career goals of young men changed? A sample proportion of young men in 2011 who say having a high­paying career is important is compared to a sample proportion measured in 1999.

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(e) A person is studied to see whether he or she can predict the results of coin flips better than chance alone.

(f) A research group wants to determine the proportion of U.S. adults who would support the impeachment of the current president.