Estimating a Population Proportion: Confidence Intervals

Sampling Distributions & Confidence Intervals: Proportion

Estimating a Population Proportion

Estimating a population proportion involves using sample data to infer the value of a proportion (p) in the entire population. Confidence intervals provide a range of plausible values for the population proportion based on the sample data and a specified confidence level.

• Population Proportion (p): The true proportion of individuals in the population with a certain characteristic (unknown).

• Sample Proportion (\( \hat{p} \)): The proportion observed in the sample, calculated as \( \hat{p} = \frac{x}{n} \), where x is the number of successes and n is the sample size.

• Confidence Interval: An interval estimate that is likely to contain the true population proportion with a specified level of confidence (e.g., 95%, 99%).

Constructing a Confidence Interval for a Population Proportion

The confidence interval for a population proportion is calculated using the following formula:

$\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• \( \hat{p} \): Sample proportion

• n: Sample size

• \( z_{\alpha/2} \): Critical value from the standard normal distribution corresponding to the desired confidence level (e.g., 1.96 for 95%, 2.576 for 99%)

Example 1: Confidence Interval for Voter Proportion

• Scenario: In a survey of 1002 people, 701 said they voted in a recent presidential election. Find a 99% confidence interval estimate for the proportion of people who say they voted.

• Given: n = 1002, x = 701, confidence level = 99% (C-Level = 0.99)

• Sample Proportion: \( \hat{p} = \frac{701}{1002} \approx 0.6996 \)

• Critical Value: For 99% confidence, \( z_{\alpha/2} = 2.576 \)

• Confidence Interval Calculation:

$0.6996 \pm 2.576 \sqrt{\frac{0.6996(1-0.6996)}{1002}}$

• Result: The interval is approximately (0.662, 0.737), or 0.662 < p < 0.737.

Example 2: Confidence Interval for Leading Digit Proportion

• Scenario: Among 784 checks from companies suspected of fraud, 61% had amounts with leading digits of 5. Construct a 95% confidence interval estimate for the proportion of such checks.

• Given: n = 784, \( \hat{p} = 0.61 \)

• Find x: \( x = 0.61 \times 784 \approx 478 \)

• Critical Value: For 95% confidence, \( z_{\alpha/2} = 1.96 \)

• Confidence Interval Calculation:

$0.61 \pm 1.96 \sqrt{\frac{0.61(1-0.61)}{784}}$

• Result: The interval is approximately (0.576, 0.644), or 0.576 < p < 0.644.

Steps for Calculating a Confidence Interval for a Proportion

1. Identify the sample size (n) and the number of successes (x).

2. Calculate the sample proportion: \( \hat{p} = \frac{x}{n} \).

3. Determine the desired confidence level and find the corresponding z-value.

4. Compute the standard error: \( \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \).

5. Calculate the margin of error: \( z_{\alpha/2} \times \text{standard error} \).

6. Construct the confidence interval: \( \hat{p} \pm \text{margin of error} \).

Interpreting Confidence Intervals

• A 95% confidence interval means that if we were to take many samples and construct a confidence interval from each, about 95% of those intervals would contain the true population proportion.

• The confidence interval provides a range of plausible values for the population proportion based on the sample data.

Summary Table: Confidence Levels and z-values

Confidence Level	z-value (\( z_{\alpha/2} \))
90%	1.645
95%	1.96
99%	2.576

Additional info: The examples use the 1-PropZInt function on a calculator, which automates the calculation of confidence intervals for proportions. The method assumes a sufficiently large sample size so that the normal approximation is valid (np > 5 and n(1-p) > 5).