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Unformatted text preview: 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.) LargeSample Confidence Intervals for Population Mean and Proportion C.I. for mean C.I. for proportion Onesided 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 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 using the sample mean x = x 1 + x 2 + + x n n For different observed values, we may have different estimates for . Which estimate is closer to the true value? No idea Instead, we may provide an interval of values of : Make this interval include the true value of 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 of a normal population. Know = 2 . 0. For a random sample of size n : X 1 , X 2 , , X n . Let us use = X . 1. What is the distribution of X ? What is the distribution of X 2 n ? 2. Find c such that P c < X 2 n < c =0.95? Find the interval of . 3. Given the observed sample n = 10: { 2,3,1,6,5,7,10,4,9,8 } . The estimate of = x = 5 . 5. Redo part 2. Purdue University Chapter7print.tex; Last Modified: July 20, 2009 (W. Sharabati) STAT511 Summer 2009 Lecture Notes 2 Definition of C.I. of Normal Mean 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 is: x 1 . 96 n , x + 1 . 96 n or with 95% confidence: x 1 . 96 n < < x + 1 . 96 n Example 7.1.2 A normal population has unknown and = 2 . 0, if n = 31 and x = 80 . 0, what is the 95% C.I.? x 1 . 96 n = 80 . 1 . 96 2 . 31 Defining Confidence Intervals Defining Confidence Intervals Purdue University Chapter7print.tex; Last Modified: July 20, 2009 (W. Sharabati) STAT511 Summer 2009 Lecture Notes 3 Interpreting a C.I. Given a 95% C.I., it is not entirely correct to say falls in the C.I. with probability 0.95 Look at the probability P 1 . 96 < X 2 n < 1 . 96 = 0 . 95, when substitute X with observed value x , no randomness left....
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 Spring '08
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