Chapter7_print

Chapter7_print - STAT511 Summer 2009 Lecture Notes 1...

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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.) Large-Sample Confidence Intervals for Population Mean and Proportion C.I. for mean 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 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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Chapter7_print - STAT511 Summer 2009 Lecture Notes 1...

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