Confidence Interval for Proportions

Definition

The formula $$ci = \hat{p} \pm z^*(\sqrt{\frac{\hat{p}(1-\hat{p})}{n}})$$ represents a confidence interval for a population proportion based on a sample proportion. This equation combines the sample proportion, the critical value from the standard normal distribution, and the standard error of the sample proportion to estimate the range within which the true population proportion is likely to fall. Understanding this term is essential for making inferences about proportions in statistics, as it helps quantify uncertainty in estimates derived from sample data.

5 Must Know Facts For Your Next Test

1. The confidence interval provides a range of values that is likely to contain the true population proportion with a specified level of confidence, often set at 95%.
2. Increasing the sample size (n) reduces the width of the confidence interval, resulting in a more precise estimate of the population proportion.
3. The critical value (z*) is determined based on the desired confidence level; for example, z* is approximately 1.96 for a 95% confidence level.
4. If the sample size is too small or if \hat{p} is too close to 0 or 1, the normal approximation may not hold well, and alternative methods should be considered.
5. The formula assumes a simple random sample and that both np̂ and n(1-p̂) are greater than 5 to ensure valid results.

Review Questions

• How does increasing the sample size affect the confidence interval for proportions?
  • Increasing the sample size directly affects the width of the confidence interval by reducing the standard error. A larger sample size leads to a smaller standard error, which in turn narrows the confidence interval. This means we can make a more precise estimate of the population proportion, as a tighter interval indicates less uncertainty about where the true proportion lies.
• Discuss why it's important to check assumptions such as np̂ and n(1-p̂) being greater than 5 when using this formula.
  • It's crucial to check that both np̂ and n(1-p̂) are greater than 5 because these conditions ensure that the sampling distribution of \hat{p} can be approximated by a normal distribution. If these conditions are not met, then using the normal approximation may lead to inaccurate or misleading results. In cases where these assumptions fail, alternative methods such as exact binomial confidence intervals should be employed for more accurate estimates.
• Evaluate how changing the critical value affects the confidence interval and interpret its implications.
  • Changing the critical value affects both the width and interpretation of the confidence interval. A higher critical value corresponds to a higher confidence level, which results in a wider interval. This means we are more certain that our interval contains the true population proportion but at the cost of precision. Conversely, using a lower critical value decreases confidence but narrows the interval, providing a more precise estimate with less certainty about containing the true proportion.

Related terms

Sample Proportion (\hat{p}): The ratio of the number of successes in a sample to the total number of observations in that sample, used as an estimate of the population proportion.

Critical Value (z*): A value from the standard normal distribution that corresponds to a specified confidence level, used to determine the margin of error in the confidence interval.

Standard Error: A measure of the variability of the sample proportion, calculated as $$\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$, reflecting how much the sample proportion is expected to fluctuate from sample to sample.