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Quantitative Methods CONFIDENCE INTERVALS Notation n = sample size, C = confidence level X̄ = sample mean s = sample standard deviation p̂ = sample proportion µ = population mean σ = population standard deviation p = population proportion Confidence interval for a mean s X̄ ± t∗ √ n where t∗ is the critical value for the t-distribution with df = n − 1. How to find t∗ : Using the t-distribution table, identify the row corresponding to df = n − 1, and the column corresponding to the confidence level (shown at the bottom). At the intersection you will find the value of t∗ . Confidence intervals for a proportion r p̂ ± z ∗ p̂(1 − p̂) n where z ∗ is the critical value for the normal distribution. How to find z ∗ : The area under the normal curve below z ∗ needs to be the confidence level C + the area of the 1−C tail, which is 1−C 2 . So, calculate the desired area C + 2 , and look inside the normal distribution table to find that number (or the closest number to it). The corresponding row and column will give you the value of z ∗ . REMARKS • If we take a large number of samples and find a 95% confidence interval for each, then the population mean (or proportion) would be contained in the interval 95% of the times. • For higher confidence levels, the interval will be wider. • The bigger the sample, the more certain we are about X (or p), and therefore the confidence interval gets smaller.