Two-Sample Test for Proportions: Fresh Take

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Two-Sample Test for Proportions: Fresh Take

Complete a two-sample [latex]z[/latex]-test for proportions from hypotheses to conclusions.
Recognize when a one-sample [latex]z[/latex]-test or a two-sample [latex]z[/latex]-test is needed to answer a research question.

A one-sample test of proportions tests a claim about a population proportion. A two-sample test of proportions tests a claim about two population proportions. When testing a claim that compares two populations, you must also check that the two populations are independent.

Let’s see if we can distinguish between these situations.

Comparing two proportions 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. Let’s begin by stating null and alternative hypotheses.

two-sample [latex]z[/latex]-test of proportions

- Null hypothesis: [latex]H_0: p_1=p_2[/latex] or [latex]H_0: p_1-p_2=0[/latex]
- Alternative hypothesis:
  - [latex]H_A: p_1\lt p_2[/latex] or [latex]H_A: p_1-p_2\lt 0[/latex]
  - [latex]H_A: p_1>p_2[/latex] or [latex]H_A: p_1-p_2>0[/latex]
  - [latex]H_A: p_1\ne p_2[/latex] or [latex]H_A: p_1-p_2\ne0[/latex]
For testing a one-sample [latex]z[/latex]-test for proportions, we require:
- Large Counts: For [latex]\hat{p}_c = \frac{x_1+x_2}{n_1+n_2}[/latex], check that: [latex]n_1\hat{p}_c \ge 10[/latex], [latex]n_2\hat{p}_c \ge 10[/latex], [latex]n_1(1-\hat{p}_c) \ge 10[/latex], and [latex]n_2(1-\hat{p}_c) \ge 10[/latex].
- Random samples/assignment: Check that the two samples are independent and random samples or that they come from randomly assigned groups in an experiment.
- 10%: Check that [latex]n_1<0.10(N_1)[/latex] and [latex]n_2<0.10(N_2)[/latex].
Calculate a test statistic.
Calculate a P-value.

Write a conclusion in context (e.g., we do/do not have convincing evidence…).

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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 [latex]200[/latex] adults given medication A still had hives [latex]30[/latex] minutes after taking the medication.
Twelve out of another random sample of [latex]200[/latex] adults given medication B still had hives [latex]30[/latex] minutes after taking the medication.

Test at a [latex]1\%[/latex] level of significance.

The problem asks for a difference in proportions, making it a test of two proportions.

Let [latex]A[/latex] and [latex]B[/latex] be the subscripts for medication [latex]A[/latex] and medication [latex]B[/latex], respectively. Then [latex]p_A[/latex] and [latex]p_B[/latex] are the desired population proportions.

Step 1: Write out the hypotheses.

[latex]H_0: p_A - p_B = 0[/latex]
[latex]H_A: p_A - p_B \ne 0[/latex]

Step 2: Check the conditions.

Our samples are random, so there is no problem there. Again, we want to determine whether the normal model is a good fit for the sampling distribution of sample proportions.

[latex]\hat{p}_c = \frac{20+12}{200+200} = 0.08[/latex]

[latex]n_1\hat{p}_c = n_2\hat{p}_c = 200(0.08) = 16 \ge 10[/latex]

[latex]n_1(1-\hat{p}_c) = n_2(1-\hat{p}_c) = 200(1-0.08) = 200(0.92) = 184 \ge 10[/latex]

Because these are all more than [latex]10[/latex], we can use the normal model to find the P-value.

Step 3: Calculate the test statistic.

Since we can use the normal model, we need to calculate the [latex]z[/latex]-test statistic for the difference in the sample proportion. Using the statistical tool, we get [latex]z= 1.47[/latex].

The observed sample difference is about [latex]1.47[/latex] standard errors above the null hypothesis value of [latex]0[/latex].

Step 4: Calculate the P-value.

Using our statistical tool, the P-value is [latex]0.1404[/latex].

Step 5: Compare P-value with significance level.

Now we compare the P-value to the level of significance, [latex]\alpha = 0.01[/latex]. In this case, the P-value of [latex]0.1404[/latex] is greater than [latex]0.01[/latex], which means we do not have enough evidence to reject the null hypothesis.

Step 6: Write conclusion in context of the problem.

At a [latex]1\%[/latex] level of significance, from the sample data, there is not sufficient evidence to conclude that there is a difference in the proportions of adult patients who did not react after [latex]30[/latex] minutes to medication [latex]A[/latex] and medication [latex]B[/latex].

Decision	Conclusion
If P-value [latex]\le\alpha[/latex], there is enough evidence to reject the null hypothesis.	At the [latex]\alpha\times[/latex]100% significance level, the data provide convincing evidence in support of the alternative hypothesis.
If P-value [latex]\gt\alpha[/latex], there is not enough evidence to reject the null hypothesis.	At the [latex]\alpha\times[/latex]100% significance level, the data do not provide convincing evidence in support of the alternative hypothesis.