lecture06 notes

# Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 06 Confidence Intervals Moo K. Chung [email protected] September 21, 2004 1. A confidence interval (CI) is an interval used to estimate the likely size of a population parameter. A confidence level is a measure of the degree of reliability of the confidence interval. Most commonly used confidence levels are the 90 % , 95 % and 99 % confidence intervals that have 0.90, 0.95 and 0.99 probabilities respectively of containing the parameter. Ex. For population parameter μ , 95 % con- fidence interval μ L , ˆ μ U ) of μ is an interval that satisfies P μ L μ ˆ μ U ) = 0 . 95 . We usually make the interval centered so that P μ L μ ) = P ( μ ˆ μ U ) = 0 . 025 . 2. Let X i N ( μ, σ 2 ) with known σ 2 and un- known μ . 95 % confidence interval for μ is ˆ μ L = ¯ x - 1 . 96 · σ/ n, ˆ μ U = ¯ x +1 . 96 · σ/ n. 3. Let X i N ( μ, σ 2 ) with known σ 2 and un- known μ . 100(1 - α )% confidence interval for μ is. ˆ μ L = ¯ x - z α/ 2 · σ/ n, ˆ μ U = ¯ x + z α/ 2 · σ/ n, where quantile z α/ 2 is given by P ( Z > z α/ 2 ) = α 2 . 4. A 95% confidence interval can be interpreted
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