Testing Hypotheses About Proportions: One-Proportion z-Test

Chapters 16 & 17: Testing Hypotheses About Proportions

Motivating Example

Hypothesis testing for proportions is a fundamental statistical method used to determine whether an observed difference in sample proportions reflects a true difference in the population or is simply due to chance variation. For example, a manufacturing process with a historical defect rate of 6% is evaluated after a process change. A random sample of 300 machines produced under the new process reveals a defect rate of 5% (15 out of 300). The question is whether this reduction is statistically significant.

• Population parameter of interest: The true proportion of defective items produced by the new process.

• Statistical question: Is the observed reduction in defect rate real, or could it be due to random sampling variation?

Fundamentals of Hypothesis Testing

Definition of Hypothesis

In statistics, a hypothesis is a statement or claim about a population parameter (a numerical characteristic of a population). Hypothesis tests are used to assess these claims.

Null and Alternative Hypotheses

• Null hypothesis ($H_0$): A statement about the value of a population parameter, typically representing no effect or no difference. For a population proportion $p$, the null hypothesis is usually $H_0: p = p_0$, where $p_0$ is a specified value.

• Alternative hypothesis ($H_A$): A statement that contradicts the null hypothesis, suggesting that the observed difference is real. It can take three forms:

  • $H_A: p \neq p_0$ (two-sided, two-tailed test)

  • $H_A: p > p_0$ (one-sided, right-tailed test)

  • $H_A: p < p_0$ (one-sided, left-tailed test)

One-Proportion z-Test

The hypothesis test for a population proportion is called a one-proportion z-test.

Procedure for Hypothesis Testing

1. Identify the population and parameter of interest.

2. Set up the hypotheses: Null ($H_0$) and Alternative ($H_A$).

3. Compute a test statistic based on the sample data.

4. Compute the P-value associated with the test statistic.

5. Draw a conclusion based on the P-value and the chosen significance level ($\alpha$).

Sampling Distribution and Test Statistic

• For large samples, the sampling distribution of the sample proportion $\hat{p}$ is approximately Normal with mean $p$ and standard deviation $\sqrt{\frac{p(1-p)}{n}}$.

• Assuming $H_0$ is true ($p = p_0$), the null model for $\hat{p}$ is: $N\left(p_0, \sqrt{\frac{p_0(1-p_0)}{n}}\right)$

• Conditions for Normal approximation: $np_0 \geq 10$ and $n(1-p_0) \geq 10$.

Calculating the Test Statistic

• The test statistic for a population proportion is the z-score: $z_{\hat{p}} = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

• This measures how many standard deviations the observed sample proportion is from the hypothesized population proportion under $H_0$.

P-Value and Statistical Significance

• The P-value is the probability, assuming $H_0$ is true, of obtaining a test statistic as extreme or more extreme than the observed value.

• For two-sided tests, the P-value is the area in both tails beyond $|z_{\hat{p}}|$.

• For one-sided tests, the P-value is the area in the relevant tail (right or left) beyond $z_{\hat{p}}$.

• A small P-value (less than $\alpha$) suggests strong evidence against $H_0$.

Decision and Conclusion

• Significance level ($\alpha$): Common values are 0.01, 0.05, and 0.10. This threshold is set before examining the data.

• If P-value $< \alpha$, reject $H_0$. The result is statistically significant.

• If P-value $\geq \alpha$, do not reject $H_0$. The result is not statistically significant.

• Never say "accept $H_0$" or "prove $H_0$ is true"; we only gather evidence for or against $H_0$.

Type I and Type II Errors

• Type I error: Rejecting $H_0$ when $H_0$ is true (false positive).

• Type II error: Failing to reject $H_0$ when $H_0$ is false (false negative).

• A small probability of both errors is desirable.

Decision	$H_0$ true	$H_A$ true
Reject $H_0$	Type I error	Correct decision
Do not reject $H_0$	Correct decision	Type II error

Example Application

• Parameter of interest: Proportion of defective items with the new process ($p$).

• Null hypothesis: $H_0: p = 0.06$

• Alternative hypothesis: $H_A: p < 0.06$ (testing for a reduction in defect rate)

• Sample data: $n = 300$, $\hat{p} = 0.05$

• Test statistic: $z_{\hat{p}} = \frac{0.05 - 0.06}{\sqrt{\frac{0.06 \times 0.94}{300}}}$

• P-value: Area to the left of the calculated $z$ in the standard normal distribution (for left-tailed test).

• Conclusion: Compare P-value to $\alpha$ (e.g., 0.10) to decide whether to reject $H_0$.

Summary of Steps in Hypothesis Testing

1. Identify the population and parameter of interest.

2. Set up the null and alternative hypotheses.

3. Compute the test statistic.

4. Compute the P-value.

5. Draw a conclusion based on the P-value and significance level.

Additional info: These notes provide a comprehensive overview of the one-proportion z-test, including the logic of hypothesis testing, calculation of test statistics, interpretation of P-values, and the importance of error types in statistical inference.