Sampling Distributions & Confidence Intervals: Proportion

Distribution of the Sample Proportion

The sample proportion, denoted as p̂, is a statistic that estimates the proportion of a population possessing a certain characteristic. Understanding its distribution is essential for making statistical inferences about population proportions.

• Definition: The sample proportion p̂ is calculated as the number of successes divided by the sample size: , where x is the number of successes and n is the sample size.

• Expected Value: The mean of the sampling distribution of p̂ is equal to the population proportion: .

• Standard Error: The standard deviation (standard error) of p̂ is: .

• Normal Approximation: The sampling distribution of p̂ is approximately normal if and .

Example: If 102 out of 350 jurors are women, . If the population proportion is , then the standard error is .

Calculating Probabilities for Sample Proportions

Probabilities involving sample proportions can be found using the normal approximation, provided the sample size is sufficiently large.

• Finding Probabilities: To find , standardize using the z-score: .

• Application: For , , , .

• Interpretation: A very low probability (e.g., 0.0000000652) suggests that such a sample proportion is extremely unlikely if the population proportion is truly 0.53.

Example: There is about a 0.0000000652 chance that 102 women or fewer would be chosen at random from this population if .

Estimating a Population Proportion

Estimating a population proportion involves using sample data to infer the value of p for the entire population.

• Point Estimate: The best point estimate for the population proportion is the sample proportion .

• Example: In a survey of 2019 adults, 1252 stated they frequently worry about their financial situation. .

Confidence Intervals for a Population Proportion

A confidence interval provides a range of plausible values for the population proportion, based on sample data and a specified level of confidence.

• Formula: The general form for a confidence interval for a proportion is:

  where is the critical value from the standard normal distribution corresponding to the desired confidence level (e.g., for 95% confidence).

• Margin of Error:

• Interpretation: "We are 95% confident that the true population proportion lies between the lower and upper bounds of the interval."

Example: For , , the 95% confidence interval is , or (0.625, 0.655).

Effect of Confidence Level and Sample Size

The width of a confidence interval depends on both the confidence level and the sample size.

• Increasing Confidence Level: Higher confidence levels (e.g., 99% vs. 95%) result in wider intervals because increases.

• Increasing Sample Size: Larger sample sizes decrease the margin of error, resulting in narrower intervals.

• Example: If the sample size increases from 2019 to 3000, the margin of error decreases, making the interval narrower.

Determining Sample Size for a Desired Margin of Error

To achieve a specific margin of error for a confidence interval, the required sample size can be calculated using the following formula:

• Formula:

• Conservative Estimate: If is unknown, use for the maximum required sample size.

Example: To estimate a proportion with and , .

Summary Table: Key Formulas for Proportion Estimation

Applications and Interpretation

• Jury Selection Example: Statistical analysis can reveal whether observed sample proportions (e.g., proportion of women jurors) are consistent with expected population proportions, helping to identify potential bias.

• Survey Example: Confidence intervals are used to estimate the proportion of adults with certain characteristics, such as financial concerns, based on sample data.

Additional info: These notes expand on handwritten calculations and examples, providing full definitions, formulas, and context for the statistical procedures involved in estimating and interpreting population proportions using sample data.