Sampling Distributions and Confidence Intervals for Proportions

9.1 The Distribution of Sample Proportions

When investigating a population proportion, the sample proportion obtained from a single sample is only one of many possible outcomes. Understanding the variability of the sample proportion across all possible samples is essential for statistical inference.

• Sample Proportion (\( \hat{p} \)): The proportion of successes in a sample, used to estimate the true population proportion (\( p \)).

• Sampling Distribution: The distribution of sample proportions from many independent samples of the same size drawn from the same population.

• Simulation: By simulating many samples, we can visualize the variability of \( \hat{p} \) and understand its distribution.

• Labeling Outcomes: For binary events, outcomes are labeled as "success" or "failure." The true proportion of successes is denoted by \( p \).

• Sampling Variability: The difference between sample proportions is expected and is referred to as sampling variability, not error.

9.2 The Sampling Distribution for Proportions

The sampling distribution of the sample proportion describes how \( \hat{p} \) varies from sample to sample. For large enough samples, this distribution is approximately Normal, centered at the true proportion \( p \).

• Mean of Sampling Distribution: \( \mu_{\hat{p}} = p \)

• Standard Deviation (Standard Error): \( SD(\hat{p}) = \sqrt{\frac{pq}{n}} \), where \( q = 1 - p \) and \( n \) is the sample size.

• Normal Model: For sufficiently large samples, the sampling distribution of \( \hat{p} \) can be modeled by a Normal distribution.

Assumptions and Conditions

• Independence Assumption: Sampled values must be independent.

• Randomization Condition: Data should come from a random sample or randomized experiment.

• 10% Condition: Sample size should be no more than 10% of the population if sampling without replacement.

• Success/Failure Condition: Both \( np \) and \( nq \) must be at least 10.

Effect of Sample Size

• The larger the sample size \( n \), the smaller the standard deviation of \( \hat{p} \).

• Smaller standard deviation leads to more precise estimates, but larger samples are more costly.

9.3 A Confidence Interval for a Proportion

A confidence interval provides a range of plausible values for the population proportion \( p \) based on the observed sample proportion \( \hat{p} \).

• Confidence Interval Formula: For a confidence level \( C \), the interval is given by:

where \( z^* \) is the critical value from the Normal model for the desired confidence level, and \( SE(\hat{p}) = \sqrt{\frac{\hat{p}\hat{q}}{n}} \).

• Interpretation: A 95% confidence interval means that, in repeated sampling, 95% of such intervals would contain the true population proportion.

• Margin of Error (ME): The maximum expected difference between the true population parameter and a sample estimate, calculated as \( z^* \times SE(\hat{p}) \).

9.4 Margin of Error: Certainty vs. Precision

The margin of error reflects the precision of the confidence interval. Increasing the confidence level increases the margin of error, making the interval wider but more certain to contain the true parameter.

• Critical Value (\( z^* \)): The number of standard errors to extend on either side of \( \hat{p} \) for a given confidence level.

• Common Confidence Levels: 90% (\( z^* = 1.645 \)), 95% (\( z^* = 1.960 \)), 99% (\( z^* = 2.576 \)).

• Trade-off: Higher confidence means a wider interval (less precision), while a narrower interval (more precision) means less confidence.

9.5 Choosing the Sample Size

To achieve a desired margin of error at a specific confidence level, the sample size must be chosen appropriately. Larger samples yield more precise estimates but require more resources.

• Sample Size Calculation: To estimate a proportion within a margin of error \( ME \) at confidence level \( C \), solve for \( n \) in:

where \( p^* \) is a guessed value for \( p \) (often 0.5 for maximum variability).

• Cost Consideration: Increasing sample size increases cost and time required for data collection.

Guided Example: Confidence Interval for a Proportion

Suppose a poll finds that 45% of a random sample of 1010 French adults are supportive of a certain action. To construct a 95% confidence interval for the true proportion:

• Check assumptions: Randomization, 10% condition, and success/failure condition are all satisfied.

• Calculate the interval using the one-proportion z-interval formula.

• Interpretation: We are 95% confident that the true proportion of all French adults who are supportive is between the calculated bounds.

Common Pitfalls and Best Practices

• Do not confuse the sampling distribution with the distribution of the sample.

• Ensure independence and adequate sample size.

• Use correct language: Do not claim certainty about the parameter or that other samples will yield the same result.

• Be aware of potential biases and violations of assumptions.

Summary of Key Concepts

• Model variation in statistics from sample to sample with a sampling distribution.

• The sampling distribution of \( \hat{p} \) is Normal for large enough samples, with mean \( p \) and standard deviation \( \sqrt{\frac{pq}{n}} \).

• Confidence intervals provide a range of plausible values for the population proportion, with a specified level of confidence.

• Margin of error depends on the confidence level and sample size.

• Check all assumptions before constructing and interpreting confidence intervals.

• Sample size affects both the certainty and precision of estimates.