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Chapter7_print - STAT511 Summer 2009 Lecture Notes 1...

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STAT511 — Summer 2009 Lecture Notes 1 Chapter 7 July 20, 2009 Statistical Inference 7.1 Basic Properties of Confidence Intervals Chapter Overview Basics Confidence Intervals (C.I.) Large-Sample Confidence Intervals for Population Mean and Proportion C.I. for mean μ C.I. for proportion One-sided intervals Intervals Based on a Normal Population Distribution for mean t distribution One sample t C.I. Confidence Intervals for the Variance and Standard Deviation of a Normal Popu- lation What is a Confidence Interval? Point Estimate vs. Confidence Interval To estimate a parameter of a population. Say μ of a normal distribution. Given the observed value x 1 , x 2 , · · · , x n of a random sample X 1 , · · · , X n . We can: Find an point estimate of μ using the sample mean ¯ x = x 1 + x 2 + ··· + x n n For different observed values, we may have different estimates for μ . Which estimate is closer to the true value? No idea Instead, we may provide an interval of values of μ : Make this interval include the true value of μ with a certain “level of confidence” (say 0.95) Narrow interval precise estimate Point estimator and error of estimator combined. Start With An Example Example 7.1.1 Want to estimate the mean μ of a normal population. Know σ = 2 . 0. For a random sample of size n : X 1 , X 2 , · · · , X n . Let us use ˆ μ = X . 1. What is the distribution of X ? What is the distribution of X - μ 2 n ? 2. Find c such that P - c < X - μ 2 n < c =0.95? Find the interval of μ . 3. Given the observed sample n = 10: { 2,3,1,6,5,7,10,4,9,8 } . The estimate of μ = ¯ x = 5 . 5. Redo part 2. Purdue University Chapter7˙print.tex; Last Modified: July 20, 2009 (W. Sharabati)
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STAT511 — Summer 2009 Lecture Notes 2 Definition of C.I. of Normal Mean μ Definition 1. After observing X 1 = x 1 , X 2 = x 2 , · · · , X n = x n . We compute the the observed sample mean ¯ x and the 95% C.I. for μ is: ¯ x - 1 . 96 σ n , ¯ x + 1 . 96 σ n or with 95% confidence: ¯ x - 1 . 96 σ n < μ < ¯ x + 1 . 96 σ n Example 7.1.2 A normal population has unknown μ and σ = 2 . 0, if n = 31 and ¯ x = 80 . 0, what is the 95% C.I.? ¯ x ± 1 . 96 σ n = 80 . 0 ± 1 . 96 2 . 0 31 Defining Confidence Intervals Defining Confidence Intervals Purdue University Chapter7˙print.tex; Last Modified: July 20, 2009 (W. Sharabati)
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STAT511 — Summer 2009 Lecture Notes 3 Interpreting a C.I. Given a 95% C.I., it is not entirely correct to say μ falls in the C.I. with probability 0.95 Look at the probability P - 1 . 96 < X - μ 2 n < 1 . 96 = 0 . 95, when substitute X with observed value ¯ x , no randomness left. A precise way to interpret C.I. is: with 95% confidence, μ falls in the interval calculated. Choosing a Different Confidence Level Example 7.1.1 We found the C.I. using P - 1 . 96 < X - μ σ n < 1 . 96 = 0 . 95, i.e., P - z 0 . 05 / 2 < X - μ σ n < z 0 . 05 / 2 = 0 . 95, if we change 0.95 to 0.90, we say the confidence level is changed to 90%.
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