Chapter 8-B

One & Two-Sample Proportion Hypothesis Testing

Introduction

Hypothesis testing for proportions is a fundamental statistical method used to determine whether the proportion of a certain characteristic in a population differs from a hypothesized value or between two populations. This guide covers both one-sample and two-sample proportion hypothesis tests, including step-by-step procedures, formulas, and practical examples.

Hypothesis Testing Steps for Proportions

General Steps

1. Hypothesize Formulation: State the null hypothesis (H0) and alternative hypothesis (Ha).

2. Prepare - Check the CLT Conditions: Ensure sample size and randomness conditions are met for the Central Limit Theorem to apply.

3. Compute (Statistics) to Compare: Calculate the test statistic (z-score) and corresponding p-value.

4. Interpret and Conclude: Compare the p-value with the significance level (α) and draw a conclusion in context.

One-Sample Proportion Hypothesis Test

• Null Hypothesis (H0): $p = p_0$

• Alternative Hypothesis (Ha): $p \neq p_0$, $p > p_0$, or $p < p_0$ (depending on the research question)

Conditions for the Test

• Sample is random.

• Sample size is large enough: $n p_0 \geq 10$ and $n(1-p_0) \geq 10$

Test Statistic Formula

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

Interpretation

• Compare the p-value to α (significance level).

• If p-value < α, reject H0; otherwise, do not reject H0.

Two-Sample Proportion Hypothesis Test

• Null Hypothesis (H0): $p_1 = p_2$

• Alternative Hypothesis (Ha): $p_1 \neq p_2$, $p_1 > p_2$, or $p_1 < p_2$

Conditions for the Test

• Both samples are random and independent.

• Sample sizes are large enough:

  • $n_1 \hat{p} \geq 10$, $n_1 (1-\hat{p}) \geq 10$

  • $n_2 \hat{p} \geq 10$, $n_2 (1-\hat{p}) \geq 10$

Pooled Proportion

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

Test Statistic Formula

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

Interpretation

• Compare the p-value to α.

• If p-value < α, reject H0; otherwise, do not reject H0.

Worked Examples

Example 1: One-Sample Proportion Test

Problem: According to a study, 25% of Canadians own a horse. In a random sample of 250 Canadians, 30% owned horses. Is there enough evidence to show that the true proportion is higher than 25%? Use α = 0.05.

• Step 1: $H_0: p = 0.25$, $H_a: p > 0.25$

• Step 2: $n p_0 = 250 \times 0.25 = 62.5 \geq 10$, $n(1-p_0) = 187.5 \geq 10$

• Step 3: $\hat{p} = 0.30$ $SD = \sqrt{\frac{0.25 \times 0.75}{250}} = 0.027$ $z = \frac{0.30 - 0.25}{0.027} = 1.85$ p-value = 0.034

• Step 4: Since 0.034 < 0.05, reject $H_0$. There is enough evidence that the proportion is higher than 25%.

Example 2: Two-Sample Proportion Test

Problem: Research compared caffeine therapy and placebo groups for infant mortality. Data:

	Therapy, $n_1$	Placebo, $n_2$	Total
Death	377	431	808
No Death	560	501	1061
Total	937	932	1869

• Step 1: $H_0: p_1 = p_2$, $H_a: p_1 \neq p_2$

• Step 2: $\alpha = 0.05$ $\hat{p}_1 = 377/937 = 0.40$ $\hat{p}_2 = 431/932 = 0.46$ $\hat{p} = (377+431)/(937+932) = 0.43$ $n_1 \hat{p} = 937 \times 0.43 = 403.91 \geq 10$ $n_2 \hat{p} = 932 \times 0.43 = 400.76 \geq 10$

• Step 3: $SD = \sqrt{0.43 \times 0.57 \left(\frac{1}{937} + \frac{1}{932}\right)} = 0.023$ $z = \frac{0.40 - 0.46}{0.023} = -2.622$ p-value = 0.004

• Step 4: Since 0.004 < 0.05, reject $H_0$. Conclusion: Caffeine therapy helps; lower proportion of babies will die.

Summary Table: Steps for Proportion Hypothesis Testing

Step	One-Sample	Two-Sample
1. Hypothesize	$H_0: p = p_0$	$H_0: p_1 = p_2$
2. Prepare (CLT Conditions)	$n p_0 \geq 10$, $n(1-p_0) \geq 10$	$n_1 \hat{p} \geq 10$, $n_2 \hat{p} \geq 10$
3. Compute Statistic	$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1-p_0)}{n}}}$	$z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$
4. Interpret	Compare p-value to α	Compare p-value to α

Additional Practice Questions (with Solutions)

• Testing claims about proportions in populations (e.g., horse ownership, product usage, environmental beliefs).

• Comparing proportions between two groups (e.g., therapy vs. placebo, male vs. female ad response).

• Each solution follows the four-step process: Hypothesize, Prepare, Compute, Interpret.

Key Terms

• Proportion ($p$): The fraction of the population with a certain characteristic.

• Sample Proportion ($\hat{p}$): The fraction in the sample with the characteristic.

• Significance Level ($\alpha$): The probability of rejecting the null hypothesis when it is true (Type I error).

• p-value: The probability, under the null hypothesis, of obtaining a result as extreme or more extreme than the observed result.

• Central Limit Theorem (CLT): Ensures the sampling distribution of the sample proportion is approximately normal for large samples.

Common Applications

• Medical studies comparing treatment effectiveness.

• Market research on product usage rates.

• Public opinion polling.

Additional info: The examples and tables provided illustrate both the calculation process and the interpretation of results in context, which is essential for hypothesis testing in statistics.