Calculate confidence intervals estimating the range where the population mean is likely to fall based on sample data. Used for statistical inference and decision making.
A confidence interval is a range in which a population parameter (mean, proportion, etc.) is estimated to fall. For example, a 95% confidence interval of [48, 52] means that if you conducted 100 sample surveys using the same method, about 95 would contain the true population mean between 48 and 52.
The confidence level represents the probability that the confidence interval contains the true parameter. Commonly 90%, 95%, or 99% is used, with 95% being most widely used. Higher confidence levels produce wider intervals, lower levels produce narrower intervals. High confidence is more certain but less precise, low confidence is more precise but less certain.
Standard error (SE) is the standard deviation of the sample mean, calculated as SE = s/√n. Larger sample sizes result in smaller standard errors. Margin of error is the radius of the confidence interval, calculated by multiplying the standard error by the Z-value. The confidence interval is expressed as sample mean ± margin of error.
Confidence intervals are calculated using CI = x̄ ± Z × (s/√n). Where x̄ is the sample mean, Z is the Z-value for the confidence level (1.96 for 95%), s is the sample standard deviation, and n is the sample size. For sample sizes less than 30 when the population is not normally distributed, use the t-distribution instead of Z.
Confidence intervals are used in various fields including opinion polls, clinical trials, quality control, and marketing research. For example, they are used to evaluate the effectiveness of new drugs, estimate average product lifespan, or predict voter support rates. Narrower confidence intervals indicate more precise estimates, while wider intervals indicate greater uncertainty.
A 95% confidence interval of [48, 52] does not mean there is a 95% probability that the population mean is in this range. The population mean is a fixed value that either is or is not in the interval. The correct interpretation is "if we repeated the survey using the same method, 95% of the confidence intervals would contain the true population mean."