Statistics Review: Confidence Intervals, Sample Size, and Proportions

Q25. Estimating the Population Proportion of U.S. Adults Taking Winter Vacations

Background

Topic: Sample Size Estimation for Population Proportion

This question tests your understanding of how to determine the minimum sample size required to estimate a population proportion with a specified level of confidence and margin of error. It also explores the impact of having a prior estimate versus no prior estimate.

Key Terms and Formulas

• Population Proportion ($p$): The fraction of the population with a certain characteristic.

• Confidence Level: The probability that the interval estimate contains the true population parameter (here, 95%).

• Margin of Error ($E$): The maximum expected difference between the true population parameter and the sample estimate (here, 5% or 0.05).

• Sample Size Formula (for proportion):

$n = \frac{z^2 \cdot p \cdot (1-p)}{E^2}$

• $n$ = minimum sample size

• $z$ = z-score for desired confidence level (for 95%, $z \approx 1.96$)

• $p$ = estimated population proportion (if unknown, use $p = 0.5$ for maximum variability)

• $E$ = margin of error (here, 0.05)

Step-by-Step Guidance

1. For part (a), since no preliminary estimate is available, use $p = 0.5$ to maximize the required sample size.

2. Identify the z-score for a 95% confidence level: $z \approx 1.96$.

3. Plug the values into the sample size formula: $n = \frac{(1.96)^2 \cdot 0.5 \cdot 0.5}{(0.05)^2}$.

4. For part (b), use the prior estimate $p = 0.32$ from the study. Plug this value into the formula: $n = \frac{(1.96)^2 \cdot 0.32 \cdot (1-0.32)}{(0.05)^2}$.

5. Compare the results from parts (a) and (b) by considering how the choice of $p$ affects the sample size.

Try solving on your own before revealing the answer!

Final Answer:

For part (a), the minimum sample size is approximately 385. For part (b), using $p = 0.32$, the minimum sample size is approximately 340.

Using a prior estimate ($p = 0.32$) results in a smaller required sample size compared to using $p = 0.5$, because the variability is lower when $p$ is farther from 0.5.