Confidence intervals of estimates are ranges calculated from sample data that estimate the likely values of an unknown population parameter. They provide a level of certainty that the true parameter falls within the calculated range.
Understanding Confidence Intervals
A confidence interval is not simply a range of plausible values; it's a range constructed with a particular confidence level. This confidence level represents the proportion of times that the interval will contain the true population parameter if the process of constructing the interval is repeated multiple times. Common confidence levels are 90%, 95%, and 99%.
Components of a Confidence Interval
A confidence interval is typically expressed as:
Estimate ± Margin of Error
Where:
- Estimate: This is the point estimate, often the sample mean or proportion, used to estimate the population parameter.
- Margin of Error: This reflects the uncertainty in the estimate and is influenced by the sample size, the variability in the sample, and the chosen confidence level.
How Confidence Intervals Work
- Select a Confidence Level: Decide on the desired confidence level (e.g., 95%).
- Calculate the Sample Statistic: Compute the point estimate from your sample data (e.g., sample mean).
- Determine the Critical Value: This value is based on the chosen confidence level and the distribution of the sample statistic (e.g., using a t-distribution or z-distribution).
- Calculate the Margin of Error: Multiply the critical value by the standard error of the sample statistic.
- Construct the Interval: Add and subtract the margin of error from the point estimate to create the confidence interval.
Interpretation
A 95% confidence interval means that if you were to repeat the sampling process many times, 95% of the calculated intervals would contain the true population parameter. It does not mean that there is a 95% probability that the true parameter lies within a specific calculated interval. The parameter is fixed; the interval varies.
Factors Affecting Confidence Interval Width
The width of a confidence interval is affected by:
- Sample Size: Larger sample sizes generally lead to narrower intervals (more precision).
- Variability: Higher variability (standard deviation) in the sample data leads to wider intervals (less precision).
- Confidence Level: Higher confidence levels (e.g., 99% vs. 90%) lead to wider intervals.
Example
Suppose you take a random sample of 100 students and find that their average height is 68 inches with a standard deviation of 3 inches. You want to construct a 95% confidence interval for the true average height of all students.
- Estimate: 68 inches (sample mean)
- Critical Value: For a 95% confidence level and a large sample size, the critical value from the z-distribution is approximately 1.96.
- Standard Error: The standard error is the standard deviation divided by the square root of the sample size: 3 / sqrt(100) = 0.3.
- Margin of Error: 1.96 * 0.3 = 0.588
- Confidence Interval: 68 ± 0.588, or (67.412, 68.588)
We can be 95% confident that the true average height of all students is between 67.412 and 68.588 inches.
Applications
Confidence intervals are widely used in various fields, including:
- Medical Research: Estimating the effectiveness of a new drug.
- Market Research: Estimating the proportion of customers who prefer a certain product.
- Political Polling: Estimating the support for a political candidate.
- Quality Control: Estimating the defect rate of a manufacturing process.
In summary, confidence intervals provide a valuable way to quantify the uncertainty associated with estimates and offer a range within which the true population parameter is likely to fall.
