Unit 3: Inference for Categorical Data: Proportions

Estimation And Hypothesis Testing

Statistical inference uses sample data to make conclusions about a population parameter. For proportions, the parameter is usually:

• $p$: one population proportion.
• $p_1-p_2$: difference between two population proportions.

A confidence interval estimates a plausible range of values for a parameter. A hypothesis test evaluates whether sample data provide convincing evidence against a null hypothesis.

On the examInference writing

For an AP free-response inference problem, the calculation is usually not enough. Name the parameter, check conditions in context, show the correct formula or calculator procedure, and finish with a conclusion that answers the original question.

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Estimators and Sampling Distributions

An estimator is a statistic used to estimate a population parameter. For categorical data, the most common estimators are:

• $\hat{p}$ for a population proportion $p$.
• $\hat{p}_1-\hat{p}_2$ for a difference in population proportions $p_1-p_2$.

A good estimator is usually unbiased and has low variability. For one sample proportion,

$$\mu_{\hat{p}}=p$$

and

$$\sigma_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}.$$

The normal approximation is appropriate when the large-counts condition is met:

$$np\ge 10 \quad \text{and} \quad n(1-p)\ge 10.$$

For two independent sample proportions,

$$\mu_{\hat{p}_1-\hat{p}_2}=p_1-p_2$$

and

$$\sigma_{\hat{p}_1-\hat{p}_2} =\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}.$$

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Confidence Intervals

A confidence interval has the form

$$\text{statistic} \pm \text{critical value}\cdot \text{standard error}.$$

The confidence level describes the long-run capture rate of the method. A 95% confidence interval does not mean there is a 95% probability that the fixed parameter is in this particular interval. It means that if we repeatedly sampled and built intervals the same way, about 95% of those intervals would contain the true parameter.

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One-Proportion z-Interval

Use a one-proportion z-interval to estimate one population proportion $p$:

$$\hat{p} \pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.$$

Conditions

1. Random sample or random assignment.
2. Independence: if sampling without replacement, $n \le 0.10N$.
3. Large counts: $n\hat{p} \ge 10$ and $n(1-\hat{p}) \ge 10$.

Common critical values:

Confidence level	$z^*$
90%	1.645
95%	1.960
99%	2.576

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Margin Of Error

The margin of error for a one-proportion interval is

$$ME = z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.$$

For planning sample size, use

$$n = \frac{(z^*)^2p^*(1-p^*)}{ME^2},$$

where $p^*$ is a planning estimate. If no estimate is given, use $p^*=0.5$ because it gives the most conservative, largest required sample size.

Always round required sample size up.

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Hypothesis Tests

A hypothesis test begins with:

• Null hypothesis $H_0$: the default claim, usually “no difference” or “equals a stated value.”
• Alternative hypothesis $H_a$: the claim we seek evidence for.

For one proportion:

$$H_0: p=p_0.$$

The alternative may be

$$H_a:p>p_0,\qquad H_a:p<p_0,\qquad \text{or}\qquad H_a:p\ne p_0.$$

The p-value is the probability, assuming $H_0$ is true, of getting a test statistic as extreme as or more extreme than the observed result in the direction of $H_a$.

Problem-solving strategy

1. Define the parameter in context.
2. State $H_0$ and $H_a$ using the parameter, not just words.
3. Check conditions using the null value when needed.
4. Compute the test statistic and p-value.
5. Decide using $\alpha$, then write the conclusion in context.

Decision rule:

• If p-value $< \alpha$, reject $H_0$.
• If p-value $\ge \alpha$, fail to reject $H_0$.

Warning

Never say “accept $H_0$”; the data may simply not be strong enough to reject it.

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One-Proportion z-Test

Use a one-proportion z-test for a claim about one population proportion:

$$z = \frac{\hat{p}-p_0}{\sqrt{p_0(1-p_0)/n}}.$$

Use $p_0$ in the standard error because the test assumes the null hypothesis is true.

Conditions

1. Random sample or random assignment.
2. Independence: if sampling without replacement, $n \le 0.10N$.
3. Large counts using the null value: $np_0 \ge 10$ and $n(1-p_0) \ge 10$.

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Two-Proportion z-Interval

Use a two-proportion z-interval to estimate $p_1-p_2$:

$$(\hat{p}_1-\hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}.$$

Conditions

1. Two random samples or random assignment to two groups.
2. Independence within and between groups.
3. If sampling without replacement, $n_1 \le 0.10N_1$ and $n_2 \le 0.10N_2$.
4. Large counts in both groups: successes and failures are each at least 10.

Interpret the interval in context: “We are ___% confident that the true difference in population proportions $p_1-p_2$ is between ___ and ___.”

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Two-Proportion z-Test

For a test of

$$H_0:p_1-p_2=0,$$

we pool the sample proportions because the null says the two population proportions are equal:

$$\hat{p}_c = \frac{x_1+x_2}{n_1+n_2}.$$

The test statistic is

$$z = \frac{(\hat{p}_1-\hat{p}_2)-0} {\sqrt{\hat{p}_c(1-\hat{p}_c)\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}.$$

Use the pooled proportion only for the hypothesis test, not for the confidence interval.

Warning

For two proportions, intervals and tests use different standard errors. Intervals use the two separate sample proportions because you are estimating the difference. Tests for $H_0:p_1-p_2=0$ use the pooled proportion because the null assumes the two population proportions are equal.

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Errors And Power

A Type I error occurs when we reject a true null hypothesis. Its probability is $\alpha$, the significance level.

A Type II error occurs when we fail to reject a false null hypothesis. Its probability is $\beta$.

Power is the probability of correctly rejecting a false null hypothesis:

$$\text{Power} = 1-\beta.$$

Power increases when:

• The true parameter is farther from the null value.
• Sample size increases.
• Significance level $\alpha$ increases.
• Variability decreases.

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Chi-Square Tests for Homogeneity and Independence

Chi-square procedures compare observed counts to expected counts. The test statistic is

$$\chi^2=\sum \frac{(O-E)^2}{E}.$$

Here $O$ is an observed count and $E$ is an expected count. Large values of $\chi^2$ indicate that the observed counts are far from what the null hypothesis predicts, so chi-square tests are right-tailed.

The chi-square distribution is right-skewed and indexed by degrees of freedom. Chi-square values are always nonnegative because the statistic is built from squared differences. As degrees of freedom increase, the distribution becomes less skewed.

Key infoObserved versus expected

Observed counts come from the data. Expected counts come from the null hypothesis. The chi-square statistic measures how far the data are from what the null model predicts.

Conditions

1. Counts come from random samples, random assignment, or a randomized process.
2. Observations are independent. If sampling without replacement, check the 10% Condition.
3. Expected counts are large enough. AP Statistics commonly uses all expected counts at least 5.

Warning

Use counts, not proportions or percentages, in the chi-square statistic.

Homogeneity

A chi-square test for homogeneity compares the distribution of one categorical variable across two or more populations or treatment groups.

• $H_0$: The category distribution is the same for all populations or treatments.
• $H_a$: At least one population or treatment has a different category distribution.

Independence

A chi-square test of independence checks whether two categorical variables are associated in one population.

• $H_0$: The two variables are independent in the population.
• $H_a$: The two variables are associated in the population.

For both homogeneity and independence, the expected count for a cell is

$$E=\frac{(\text{row total})(\text{column total})}{\text{grand total}}.$$

The degrees of freedom are

$$df=(r-1)(c-1),$$

where $r$ is the number of rows and $c$ is the number of columns.

Problem-solving strategy

• Use homogeneity when there are separate samples or treatment groups and one categorical response.
• Use independence when there is one sample and two categorical variables measured on each individual.

Contributions

Each cell’s contribution is

$$\frac{(O-E)^2}{E}.$$

The largest contributions show which cells are most responsible for the overall chi-square statistic. A cell with $O>E$ occurred more often than expected under the null; a cell with $O<E$ occurred less often than expected.

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Calculator Notes

Common calculator tools:

• 1-PropZInt: one-proportion confidence interval.
• 1-PropZTest: one-proportion hypothesis test.
• 2-PropZInt: two-proportion confidence interval.
• 2-PropZTest: two-proportion hypothesis test.
• X2-Test: chi-square test for homogeneity or independence using a matrix of observed counts.

Calculator output does not replace communication. You still need hypotheses, conditions, p-value or interval, and a conclusion in context.

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Working Checklist

Working checklist

1. Identify the parameter: $p$ or $p_1-p_2$.
2. Choose interval or test.
3. Check random, independence, and large-count conditions.
4. Use the correct standard error: null value for tests, sample value for intervals.
5. Compute the interval or p-value.
6. Write a conclusion in context.

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Key Equations

Key equations

Idea	Equation
One-proportion interval	$\hat{p}\pm z^*\sqrt{\hat{p}(1-\hat{p})/n}$
One-proportion test	$z=(\hat{p}-p_0)/\sqrt{p_0(1-p_0)/n}$
Two-proportion interval	$(\hat{p}_1-\hat{p}_2)\pm z^*\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$
Pooled proportion	$\hat{p}_c=(x_1+x_2)/(n_1+n_2)$
Two-proportion test	$z=\frac{(\hat{p}_1-\hat{p}_2)}{\sqrt{\hat{p}_c(1-\hat{p}_c)(1/n_1+1/n_2)}}$
Chi-square statistic	$\chi^2=\sum\dfrac{(O-E)^2}{E}$
Expected count	$E=\dfrac{(\text{row total})(\text{column total})}{\text{grand total}}$