Confidence Intervals for Proportions

Chapter 15: Confidence Intervals for Proportions

Introduction to Confidence Intervals for Proportions

Confidence intervals provide a range of plausible values for a population parameter, such as a proportion, based on sample data. This chapter focuses on constructing and interpreting confidence intervals for population proportions.

• Population proportion (p): The true proportion of interest in the population.

• Sample proportion (\(\hat{p}\)): The proportion observed in the sample, used to estimate \(p\).

If the sample size \(n\) is sufficiently large (\(np \geq 10\) and \(n(1-p) \geq 10\)), the sampling distribution of \(\hat{p}\) is approximately normal:

$$ \hat{p} \sim N\left(p, \frac{p(1-p)}{n}\right)$$

The standard error (SE) of \(\hat{p}\) is:

$$ SE(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

The 68-95-99.7 Rule and Confidence Intervals

The empirical rule (68-95-99.7 rule) helps us understand how sample proportions vary:

• About 68% of sample proportions fall within \(\hat{p} \pm SE(\hat{p})\) of \(p\).

• About 95% fall within \(\hat{p} \pm 2SE(\hat{p})\) of \(p\).

• About 99.7% fall within \(\hat{p} \pm 3SE(\hat{p})\) of \(p\).

These intervals are called confidence intervals for the population proportion.

Confidence Intervals for the Population Proportion

• General formula for a confidence interval (CI):

$$ \hat{p} \pm z^* SE(\hat{p})$$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

• \(z^*\) is the critical value from the standard normal distribution, determined by the desired confidence level.

Confidence Level (C)

• The probability (expressed as a percentage) that a confidence interval encloses the population proportion \(p\).

• Common confidence levels: 90%, 95%, 99%.

The confidence level determines the critical value \(z^*\):

$$ \text{Upper tail area} = \frac{100\% - \text{confidence level}}{2}$$

For example, for a 90% CI, \(z^* = 1.645\).

Margin of Error (ME)

• Measures the width of the confidence interval and the precision of the estimate.

$$ ME = z^* SE(\hat{p})$$

A larger margin of error means a wider interval (more likely to contain \(p\)), but less precision.

Assumptions and Conditions for Constructing a Confidence Interval

• The sample is randomly drawn from the population (to avoid bias).

• Sampled values are independent (sample size is a small fraction of the population size).

• Sample size is sufficiently large: \(np \geq 10\) and \(nq \geq 10\), where \(q = 1-p\).

Properties of a Confidence Interval

• Centered at the sample proportion \(\hat{p}\).

• Width increases with higher confidence level (lower precision).

• Width decreases as sample size increases (greater precision).

Interpretation of a Confidence Interval

• Over many repeated samples, a proportion \(C\)% of confidence intervals constructed will contain the true population proportion \(p\).

• We say we are \(C\)% confident that the true proportion falls within the interval obtained from our sample.

Important Notes about Confidence Intervals

• Confidence intervals are constructed for parameters (like \(p\)), not statistics (like \(\hat{p}\)).

• The population proportion \(p\) is fixed; the sample proportion \(\hat{p}\) and the confidence interval are random due to sampling variability.

• For a specific interval (e.g., (0.50, 0.65)), we are 95% confident it contains \(p\), but \(p\) either is or is not in the interval—we do not assign a probability to this one interval.

Sample Size Determination for Estimating a Population Proportion

To achieve a desired margin of error (ME) at a given confidence level, the required sample size \(n\) is:

$$ n = \frac{(z^*)^2 \hat{p}(1-\hat{p})}{ME^2}$$

• If prior estimates of \(\hat{p}\) are available, use them.

• If no prior estimates, use \(\hat{p} = 0.5\) (maximizes the required sample size).

Example Exercise

• A university dean randomly selects 200 students and finds that 92 receive financial aid.

• (a) Construct a 95% confidence interval for the true proportion of students who receive financial aid.

• (b) Interpret the confidence interval.

• (c) If the sample size is tripled, the width of the 95% confidence interval will decrease (since standard error decreases as \(n\) increases).

• (d) To estimate the proportion within 0.05 with 95% confidence, calculate the required sample size using the formula above.

• (e) If no prior information is available, use \(\hat{p} = 0.5\) for the calculation.

Appendix: Maximizing Sample Size

To ensure the margin of error does not exceed a specified value, use \(\hat{p} = 0.5\) when no prior information is available, as this maximizes \(\hat{p}(1-\hat{p})\) and thus the required sample size.