Inference for a Population Proportion: Confidence Intervals and Hypothesis Testing

Inference for a Population Proportion

Population Proportion and Sample Proportion

In statistics, we often wish to estimate or test the proportion of a population that possesses a certain attribute. This is fundamental in survey analysis, quality control, and many other fields.

• Population proportion, p: The proportion (percentage) of the entire population that has the specified attribute.

• Sample proportion, \hat{p}: The proportion (percentage) of a sample from the population that has the specified attribute.

Sampling Distribution of the Sample Proportion

When we repeatedly take random samples of size n from a population and calculate the sample proportion \hat{p} for each, the distribution of these sample proportions forms the sampling distribution of \hat{p}. For large enough sample sizes, this distribution is approximately normal due to the Central Limit Theorem.

• Mean: p

• Standard deviation: $\sqrt{\frac{p(1-p)}{n}}$

• Normal approximation: Valid when both $np \geq 5$ and $n(1-p) \geq 5$.

Confidence Intervals for a Population Proportion

A confidence interval (CI) provides a range of plausible values for the population proportion p based on the sample data. The most common method is the one-proportion z-interval.

• Margin of error, E: The maximum likely difference between the sample proportion and the true population proportion at a given confidence level.

• Formula for CI: $\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Interpretation: We are (1-α)% confident that the true population proportion lies within this interval.

One-Proportion z-Interval Procedure

To construct a confidence interval for a population proportion, follow these steps:

1. For a confidence level of 1 − α, use a z-table to find $z_{\alpha/2}$.

2. Compute the confidence interval: $\hat{p} - z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ to $\hat{p} + z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.

3. Interpret the confidence interval in the context of the problem.

Common z-Values for Confidence Intervals

The value of $z_{\alpha/2}$ depends on the desired confidence level:

z0.10	z0.05	z0.025	z0.01	z0.005
1.282	1.645	1.960	2.326	2.576

Determining Sample Size for a Desired Margin of Error

To achieve a specific margin of error E for a confidence interval, the required sample size n can be calculated as:

• $n = \hat{p}(1-\hat{p}) \left( \frac{z_{\alpha/2}}{E} \right)^2 \leq \frac{1}{4} \left( \frac{z_{\alpha/2}}{E} \right)^2$

• If no prior estimate for \hat{p} is available, use 0.5 for maximum variability.

Example: Playing Hooky From Work

A poll of 1010 U.S. employees asked if they "play hooky" (call in sick at least once a year for relaxation). 202 responded "yes." This data can be used to construct a confidence interval for the proportion of employees who play hooky.

One-Proportion z-Test: Hypothesis Testing for a Population Proportion

Hypothesis testing allows us to assess claims about a population proportion. The one-proportion z-test is used when the sample size is large enough for the normal approximation to be valid.

• Null hypothesis, H0: p = p0

• Alternative hypothesis, Ha: p ≠ p0 (two-tailed), p < p0 (left-tailed), or p > p0 (right-tailed)

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

Critical-Value and P-Value Approaches

There are two main approaches to making a decision in hypothesis testing:

• Critical-value approach: Compare the test statistic to critical values from the z-table. If the statistic falls in the rejection region, reject H0.

• P-value approach: Find the probability (P-value) of observing a test statistic as extreme as, or more extreme than, the observed value. If P ≤ α, reject H0.

Example: Quality Control in Manufacturing

Suppose a manufacturer wants to ensure that fewer than 5% of batteries are defective. In a random sample of 300 batteries, 10 are found defective. We can use a one-proportion z-test to determine if there is sufficient evidence to conclude the defective rate is less than 0.05 at α = 0.01.

Summary Table: Key Formulas and Concepts

Concept	Formula	Notes
Sample Proportion	$\hat{p} = \frac{x}{n}$	x = number with attribute, n = sample size
Standard Error	$\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$	Used in CI and z-test
Confidence Interval	$\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$	1-α confidence level
z-Test Statistic	$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$	Test H0: p = p0
Sample Size for Margin E	$n = \hat{p}(1-\hat{p}) \left( \frac{z_{\alpha/2}}{E} \right)^2$	Use \hat{p} = 0.5 if unknown