Sampling Distribution Models and Confidence Intervals for Proportions: Sample Size Determination

Sampling Distribution Models and Confidence Intervals for Proportions

Choosing the Sample Size

Determining the appropriate sample size is crucial for constructing confidence intervals with a desired margin of error (ME) and confidence level. The sample size affects the precision of the estimate and the reliability of statistical conclusions.

• Population Proportion Estimate (\( \hat{p} \)): If a reasonable estimate is available from prior studies or pilot data, use it. Otherwise, use \( \hat{p} = 0.5 \) for a conservative (largest) sample size.

• Sample Size Formula: To achieve a specified margin of error (ME) at a given confidence level (with critical value \( z^* \)), use: $n = \frac{(z^*)^2 \hat{p} (1-\hat{p})}{ME^2}$

• Conservative Approach: Using \( \hat{p} = 0.5 \) maximizes \( \hat{p}(1-\hat{p}) \), ensuring the sample size is large enough for any true proportion.

Example: Determining Sample Size for a Survey

Suppose an initial poll finds 65% believe global warming is due to human activity. For a follow-up survey, what sample size is needed for a 95% confidence interval with ME < ±2%?

• Given: \( \hat{p} = 0.65, \hat{q} = 0.35, z^* = 1.96, ME = 0.02 \)

• Calculation: $n = \frac{(1.96)^2 (0.65)(0.35)}{(0.02)^2} \approx 2185$

• At least 2185 respondents are needed.

Conservative Sample Size Calculation

To ensure ME < 3% at 95% confidence, using \( \hat{p} = 0.5 \):

• Given: \( z^* = 1.96, ME = 0.03 \)

• Calculation: $n = \frac{(1.96)^2 (0.5)(0.5)}{(0.03)^2} \approx 1068$

• Always round up to the next whole number.

Example: Rare Event Sample Size

A pilot study shows 0.5% of credit card offers result in sign-up. To estimate the true rate within 0.1% (ME = 0.001) at 95% confidence:

• Given: \( \hat{p} = 0.005, \hat{q} = 0.995, z^* = 1.96, ME = 0.001 \)

• Calculation: $n = \frac{(1.96)^2 (0.005)(0.995)}{(0.001)^2} \approx 19,112$

• Using \( \hat{p} = 0.5 \) would yield an unreasonably large sample size (\( n \approx 960,400 \)).

Sample Size and Standard Deviation

Increasing the sample size reduces the standard deviation of the sampling distribution, improving estimate precision.

• Standard Deviation Formula: $SD(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• To halve the standard deviation, multiply the sample size by 4.

• To reduce the standard deviation by a factor of 10, multiply the sample size by 100.

• Larger samples are more precise but may be costly or time-consuming.

Typical Poll Sizes

Most widely reported polls use a margin of error of ±3% at 95% confidence. For any population greater than 10,680, a sample of about 1068 is sufficient for this margin of error.

• Formula: $n = \frac{(1.96)^2 (0.5)(0.5)}{(0.03)^2} \approx 1068$

• For a margin of error of 1% (ME = 0.01) and \( \hat{p} = 0.7 \): $n = \frac{(1.96)^2 (0.7)(0.3)}{(0.01)^2} \approx 8068$

• Reducing ME from 3% to 1% requires a much larger sample.

Key Terms and Concepts

• Critical Value (z*): The number of standard errors to move away from the mean of the sampling distribution to correspond to the specified confidence level.

• Margin of Error (ME): The maximum expected difference between the observed sample proportion and the true population proportion at a given confidence level.

• Confidence Interval: An interval estimate, computed from the sample data, that is likely to contain the true population parameter.

Common Misconceptions and Pitfalls

• Treat the entire confidence interval equally; the center is not necessarily more plausible than the edges.

• Beware of margins of error that are too large to be useful.

• Watch out for biased sampling; it produces unreliable confidence intervals.

• Ensure independence and randomization in sample design.

• Do not assume the Normal model always applies; check assumptions and conditions.

• Confidence intervals are about the parameter, not the observed statistic.

• Accept that confidence is not certainty; a 95% confidence interval means you are not 100% sure.