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Chapter 7 - Chapter 7.Statistical Intervals based on a...

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Chapter 7.Statistical Intervals based on a single sample Statistical Interval The point estimate is a single number (accurate) - does not provide information about precision and reliability of estimation - report an entire interval of plausible values - an interval estimate Accuracy and Precision A Confidence Interval - first select a confidence level, which is a measure of reliability of the interval Precision of an interval estimate is conveyed by the width of the confidence interval 7.1 Basic Properties of Confidence Intervals 1 2 , , , n X X X L : rs from 2 ( , ) N μ σ : unknown, : known μ σ 2 : ( , ) X N n σ μ 1
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: (0,1) / X Z N n μ σ - = ( 1. 96 1. 96) 0. 95 P Z - < < = ( 1. 96 1. 96) 0. 95 / X P n μ σ - - < < = ( 1. 96 / 1. 96 / ) 0. 95 P X n X n σ μ σ - < < + = Random Interval – interval whose end points are rv’s ( 1. 96 / , 1. 96 / ) X n X n σ σ - + 1. 96 / X n σ ± The probability that this random interval will cover the true mean μ is 95% Def) 95% Confidence Interval for μ ( 1. 96 / , 1. 96 / ) x n x n σ σ - + 1. 96 / x n σ ± Ex 7.2 : 2. 0, 31, 80. 0 n x σ = = = 95% CI : 1. 96 / 80. 0 (1. 96) 2 / 31 80. 0 0. 7 (79. 3, 80. 7) x n σ ± = ± = ± = We are confident at 95% level that 79. 3 80. 7 μ < < The interval estimate 1. 96 / X n σ ± is no longer random interval 2
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- “The probability that the random interval will cover the true mean μ is 95%” and “We are confident at 95% level that 79. 3 80. 7 μ < < are incorrect statements Interpreting a confidence interval - long-run relative frequency
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