A Confidence Interval is a type of interval estimate, derived from the Statistics field, used to estimate the true but unknown parameter of a population. Here are the key points about confidence intervals:
Definition
- A Confidence Interval provides a range of values, derived from sample data, that is likely to contain the true population parameter with a certain level of confidence, usually expressed as a percentage (e.g., 95% or 99%).
- It does not imply that the true parameter value has a 95% chance of falling within the interval; rather, it means that if the same population is sampled repeatedly, about 95% of the intervals constructed would contain the true parameter.
Formula and Calculation
The formula for a confidence interval for a population mean when the population standard deviation is known is:
CI = X̄ ± Z*(σ/√n)
- X̄: Sample mean
- Z: Z-value corresponding to the desired confidence level (e.g., 1.96 for 95% confidence)
- σ: Population standard deviation
- n: Sample size
When the population standard deviation is unknown, the t-distribution is used, and the formula becomes:
CI = X̄ ± t*(s/√n)
- s: Sample standard deviation
- t: t-value from the t-distribution table, which depends on the sample size and confidence level
History and Development
- The concept of confidence intervals was first introduced by Jerzy Neyman and Egon Pearson in the 1930s. They developed the theory of hypothesis testing and interval estimation as part of the Neyman-Pearson Lemma.
- The term "confidence" was chosen to reflect the frequentist interpretation of probability, emphasizing the long-run frequency of intervals containing the true parameter.
Interpretation and Use
- Confidence intervals are widely used in statistical inference to express the precision of sample estimates. They provide a range within which the true value is likely to lie.
- They are particularly useful in fields like medical research, where they help in understanding the effectiveness of treatments or the prevalence of diseases with a quantifiable level of certainty.
Limitations
- Confidence intervals assume that the sample data is drawn from a normally distributed population or that the sample size is large enough for the Central Limit Theorem to apply.
- They can be misleading if not interpreted correctly, especially with small sample sizes where the intervals might be too wide to be informative.
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