AP Statistics 3.9-3.10: Comparing Two Proportions

Two-independent-population-proportions test

A hypothesis test used to determine whether two independent population proportions differ. It compares sample proportions using an approximate normal distribution for their difference.

What parameter is compared in a two-proportion hypothesis test?

The population proportions $p_A$ and $p_B$ are compared, usually through the difference $p_A-p_B$. The sample difference $\hat p_A-\hat p_B$ estimates this population difference.

What conditions should be checked before using a two-proportion $z$ test?

The samples should be independent simple random samples, each population should be much larger than its sample, and each sample should have at least five expected successes and five expected failures under the testing model.

Why must the populations be much larger than the samples in a two-proportion test?

A population should generally be at least 10 or 20 times the sample size so that sampling observations do not substantially change the population composition or violate independence.

How are sample proportions calculated for two independent samples?

For group $A$, $\hat p_A=\frac{x_A}{n_A}$; for group $B$, $\hat p_B=\frac{x_B}{n_B}$. Here $x$ is the number of successes and $n$ is the sample size.

What is the usual null hypothesis for comparing two population proportions?

The usual null hypothesis states that the population proportions are equal: $H_0:p_A=p_B$, equivalently $H_0:p_A-p_B=0$.

How should the alternative hypothesis be chosen for a two-proportion test?

It should reflect the research question: $H_a:p_A\ne p_B$ for any difference, $H_a:p_A<p_B$ for a smaller proportion, or $H_a:p_A>p_B$ for a larger proportion.

How do the words in a research question determine the tail of a two-proportion test?

Words such as "different" or "a difference" indicate a two-tailed test. "Less than" indicates a left-tailed test, and "greater than," "more," or "higher" indicates a right-tailed test.

Pooled proportion

The pooled proportion combines the successes from both samples under the null assumption that the population proportions are equal: $\hat p_c=\frac{x_A+x_B}{n_A+n_B}$.

Why is a pooled proportion used in the two-proportion $z$ test?

Under $H_0:p_A=p_B$, both samples are treated as estimates of one common population proportion. Pooling the successes provides the null-model estimate needed for the standard error.

What is the standard error for the difference between two proportions under the null hypothesis?

Using the pooled proportion, the null standard error is $SE_0=\sqrt{\hat p_c(1-\hat p_c)\left(\frac{1}{n_A}+\frac{1}{n_B}\right)}$.

What is the approximate sampling distribution of $\hat p_A-\hat p_B$ under $H_0$?

If the conditions are satisfied, $\hat p_A-\hat p_B$ is approximately normal with mean $0$ and variance $\hat p_c(1-\hat p_c)\left(\frac{1}{n_A}+\frac{1}{n_B}\right)$.

What is the test statistic for comparing two independent population proportions?

$z=\frac{(\hat p_A-\hat p_B)-(p_A-p_B)_0}{\sqrt{\hat p_c(1-\hat p_c)\left(\frac{1}{n_A}+\frac{1}{n_B}\right)}}$. For the usual null hypothesis, $(p_A-p_B)_0=0$.

Why does the two-proportion test statistic use a pooled standard error instead of separate sample proportions?

The test statistic models the data assuming the null hypothesis of equal population proportions is true. Therefore, the standard error must be based on the common pooled estimate rather than separate estimates.

What does a positive value of $z$ mean when the statistic is based on $\hat p_A-\hat p_B$?

It means the observed difference is above the null value, so the sample proportion for group $A$ exceeds that for group $B$. Whether this supports the alternative depends on the direction of $H_a$.

What does a p-value represent in a two-proportion hypothesis test?

It is the probability, assuming $H_0$ is true, of obtaining a test statistic at least as extreme as the observed one in the direction specified by the alternative hypothesis.

How is the p-value determined for a two-tailed two-proportion test?

For $H_a:p_A\ne p_B$, the p-value includes outcomes at least as far from zero in both tails of the null normal distribution. For a symmetric $z$ distribution, it is $2P(Z\ge |z|)$.

How is the p-value determined for left- and right-tailed two-proportion tests?

For $H_a:p_A<p_B$, use the area to the left of the observed $z$. For $H_a:p_A>p_B$, use the area to the right.

How do you make a decision using a p-value and significance level?

Reject $H_0$ when $p\text{-value}<\alpha$. Otherwise, do not reject $H_0$; this means the data do not provide sufficiently strong evidence against the null hypothesis at level $\alpha$.

What is the correct conclusion when a two-proportion test does not reject $H_0$?

State that there is not sufficient evidence to conclude the directional or nondirectional claim in $H_a$. Do not conclude that the two population proportions are proven equal.

What is the correct conclusion when a two-proportion test rejects $H_0$?

State that there is sufficient evidence, at the chosen significance level, to support the claim in the alternative hypothesis about the population proportions.

Statistical significance level $\alpha$

The chosen maximum probability of a Type I error: incorrectly rejecting a true null hypothesis. Common values include $0.05$ and $0.01$.

Type I error in a two-proportion hypothesis test

A Type I error occurs when the test rejects $H_0$ even though the two population proportions are actually equal. Its probability is controlled by the significance level, $\alpha=P(\text{reject }H_0\mid H_0\text{ true})$.

Type II error in a two-proportion hypothesis test

A Type II error occurs when the test fails to reject $H_0$ even though the population proportions really differ in the direction of the alternative. Its probability is commonly denoted $\beta$.

Power of a two-proportion hypothesis test

Power is the probability of correctly rejecting a false null hypothesis. It equals $1-\beta$ and generally increases with larger samples, larger true differences, and a larger significance level.

Critical region

The set of test-statistic values that leads to rejection of $H_0$. Its location depends on whether the alternative is left-tailed, right-tailed, or two-tailed, and its size is determined by $\alpha$.

How does sample size affect the two-proportion $z$ statistic?

Larger sample sizes reduce the standard error $\sqrt{\hat p_c(1-\hat p_c)(1/n_A+1/n_B)}$. Thus, the same observed difference can produce a larger absolute $z$ value and stronger evidence against $H_0$.

Why can two noticeably different sample proportions fail to produce statistical significance?

The observed difference may be plausible from random sampling variation, especially when samples are small or the standard error is large. Statistical significance depends on the difference relative to its standard error, not on the difference alone.

In a medication study, 20 of 200 patients in group A and 12 of 200 in group B still have symptoms. What are the sample proportions and observed difference?

$\hat p_A=20/200=0.10$, $\hat p_B=12/200=0.06$, and $\hat p_A-\hat p_B=0.04$.

For the medication study with 20 successes out of 200 in group A and 12 out of 200 in group B, what is the pooled proportion?

$\hat p_c=\frac{20+12}{200+200}=0.08$, so $1-\hat p_c=0.92$.

For the medication study, what conclusion follows from a two-tailed p-value of $0.1404$ at $\alpha=0.01$?

Because $0.1404>0.01$, do not reject $H_0$. There is not sufficient evidence at the 1% level to conclude that the population proportions of patients still having symptoms differ between the medications.

How should a claim that one population proportion is lower than another be tested?

Define the difference in the claimed order, such as $p_F-p_M$, and use $H_a:p_F-p_M<0$. This is a left-tailed two-proportion $z$ test.

How should a claim that one population proportion is higher than another be tested?

Define the difference in the claimed order, such as $p_W-p_A$, and use $H_a:p_W-p_A>0$. This is a right-tailed two-proportion $z$ test.

A study reports a right-tailed p-value of $0.0077$ at $\alpha=0.05$. What is the statistical decision and interpretation?

Reject $H_0$ because $0.0077<0.05$. The data provide sufficient evidence at the 5% level that the first population has a greater proportion than the second.

Why is a two-proportion test a $z$ test rather than a $t$ test?

For sufficiently large binomial samples, the difference between sample proportions has an approximately normal sampling distribution. Its standardized statistic is therefore compared with the standard normal distribution.

What population characteristic is tested, rather than a sample characteristic?

A hypothesis test makes a claim about population parameters, such as $p_A$ and $p_B$. The sample proportions are used as evidence about those unknown population quantities.

What does it mean to condition a p-value on the null hypothesis?

The p-value is calculated under the assumption that $H_0$ is true. It is not the probability that $H_0$ is true or false.

Why is failing to reject $H_0$ not the same as accepting $H_0$?

A nonsignificant result means the sample evidence is insufficient to reject the null at the selected $\alpha$ level. It does not establish that the null hypothesis is certainly true.

How do critical-value and p-value methods compare?

Both methods make the same decision when applied correctly. The critical-value method compares the observed $z$ with rejection cutoffs, whereas the p-value method compares the probability of results at least as extreme with $\alpha$.

What is the sampling distribution of $\hat p_A-\hat p_B$ for two independent samples?

When the samples are independent and the large-count conditions are satisfied, $\hat p_A-\hat p_B$ is approximately normal with mean $p_A-p_B$ and standard deviation $\sqrt{\frac{p_A(1-p_A)}{n_A}+\frac{p_B(1-p_B)}{n_B}}$.

Why is the mean of the sampling distribution of $\hat p_A-\hat p_B$ equal to $p_A-p_B$?

Because $E(\hat p_A)=p_A$ and $E(\hat p_B)=p_B$, and the expected value of a difference is the difference of the expected values: $E(\hat p_A-\hat p_B)=p_A-p_B$.

What conditions should be checked before constructing a two-proportion $z$ confidence interval?

The two samples should be independent random samples, each population should be at least about 10 times its sample size when sampling without replacement, and each sample should contain at least 10 successes and 10 failures.

What standard error is used for a confidence interval for $p_A-p_B$?

Use the separate sample proportions rather than a pooled proportion: $SE=\sqrt{\frac{\hat p_A(1-\hat p_A)}{n_A}+\frac{\hat p_B(1-\hat p_B)}{n_B}}$.

What is the formula for a two-proportion $z$ confidence interval?

A confidence interval for $p_A-p_B$ is $(\hat p_A-\hat p_B)\pm z^*\sqrt{\frac{\hat p_A(1-\hat p_A)}{n_A}+\frac{\hat p_B(1-\hat p_B)}{n_B}}$, where $z^*$ is the standard-normal critical value for the chosen confidence level.

Why is the standard error for a two-proportion confidence interval not pooled?

A confidence interval estimates the actual difference between two population proportions without assuming $p_A=p_B$. Therefore, it uses each sample proportion to estimate its own population variability.

How should a two-proportion confidence interval be interpreted?

A confidence interval gives a range of plausible values for the difference $p_A-p_B$. In repeated sampling, the stated percentage of intervals constructed by this method would capture the true difference.

How does a two-proportion confidence interval indicate whether the population proportions differ?

If the interval for $p_A-p_B$ contains $0$, a difference of zero is plausible. If the interval excludes $0$, the data provide evidence of a difference at the corresponding two-sided significance level.

How does the order of subtraction affect a confidence interval for two proportions?

Switching from $p_A-p_B$ to $p_B-p_A$ changes every value in the interval to its opposite. The interval's interpretation must use the same order as the subtraction.

How do confidence level and sample size affect the width of a two-proportion confidence interval?

A higher confidence level increases $z^*$ and produces a wider interval. Larger sample sizes reduce the standard error and generally produce a narrower interval.