Understanding Confidence Intervals

• A confidence interval is a range of values, calculated from sample data, that is likely to contain an unknown population parameter
• It is associated with a confidence level that specifies the probability that the interval will capture the true parameter value in repeated samples

Definition of Confidence Interval

• A confidence interval provides a range of plausible values for an unknown population parameter
• Instead of providing a single point estimate, it gives an interval within which the parameter is expected to lie
• The confidence level associated with the interval reflects the percentage of repeated samples that would produce intervals containing the true parameter

Calculating Confidence Intervals

• The general formula for a confidence interval is: Point Estimate ± (Critical Value × Standard Error)
• The point estimate is the sample statistic used to estimate the population parameter
• The critical value is a value from a standard distribution (e.g., Z or t) that corresponds to the desired confidence level
• The standard error is the standard deviation of the sample statistic
• For population proportions, the point estimate is the sample proportion (p̂)
• The standard error is calculated as √((p̂(1-p̂))/n), where n is the sample size
• The critical value for a proportion is a Z-score, found using a standard normal distribution table or calculator

Sample Size Determination

• To calculate a confidence interval, one has to determine the required sample size
• The required sample size depends on the desired margin of error, confidence level, and estimated population proportion
• For proportions, the sample size (n) can be calculated using: n = (Z*^2 * p̂(1-p̂)) / E^2
• Z* is the critical value corresponding to the desired confidence level
• p̂ is the estimated sample proportion (use 0.5 if unknown, as it maximizes the required sample size)
• E is the desired margin of error
• When determining the sample size, always round up to the nearest whole number to ensure the desired margin of error is achieved

Interpretation of Confidence Intervals

• A confidence interval is interpreted in terms of the population parameter
• For example, a 95% confidence interval means that if we were to take many samples and construct confidence intervals from each sample, about 95% of those intervals would contain the true population proportion
• It is incorrect to say that there is a 95% chance that the true population proportion falls within a specific calculated interval because the population proportion is fixed, and the interval varies with each sample
• The confidence level reflects the reliability of the estimation procedure rather than the certainty about a specific interval