JMF-7&8-07 - CH Reilly UCF - IEMS - STA 3032 1...

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Unformatted text preview: CH Reilly UCF - IEMS - STA 3032 1 Inferences About Means Chapter 7 Notes Set 7 Spring 2007 CH Reilly UCF - IEMS - STA 3032 2 Estimation Two kinds of estimates that we may make: Point estimate a single value. Interval estimate a range of values. Although we most typically think of point estimates, there are times when we will be interested in estimating a range of reasonable values for some population parameter. Ex: The average number of runs scored in a baseball game is between 4 and 12. CH Reilly UCF - IEMS - STA 3032 3 Point estimation . Similarly, . for estimator point a is because . for estimator point a is parameter. a is 2 2 S X X = = CH Reilly UCF - IEMS - STA 3032 4 Unbiasedness 2 2 2 1 2 2 1 ) ( E since unbiased is . ) ( E since unbiased is 1 ) ( 1 . ) ( E if unbiased is = =-- = = = = = S S X X n X X S X n X n i i n i i CH Reilly UCF - IEMS - STA 3032 5 Efficiency ) ( Var ) ( Var if than efficient more is . estimators unbiased are , that Suppose 2 1 2 1 2 1 CH Reilly UCF - IEMS - STA 3032 6 Example Suppose that we have n random values from a Poisson distribution with (population) mean . Which of the following is the best point estimator for ? Sample mean. First observation divided by n . Mean of first two observations. Source: Mendenhall and Sincich (1994), Problem 8.2. CH Reilly UCF - IEMS - STA 3032 7 Example (contd.) ( 29 = = + = + = = = = = = = = = = = ) 2 ( 2 1 ) ( E ) ( E 2 1 2 E ) ( E ) ( E 1 E ) ( E ) ( 1 ) ( E 1 1 E ) ( E ) ( E 2 1 2 1 3 1 1 2 1 1 1 X X X X n X n n X n n X n X n X n i i n i i CH Reilly UCF - IEMS - STA 3032 8 Example (contd.) ( 29 . 2 as long as ) ( Var ) ( Var 2 ) 2 ( 4 1 ) ( Var ) ( Var 4 1 2 Var ) ( Var ) ( 1 ) ( Var 1 1 Var ) ( Var 3 1 2 1 2 1 3 2 1 2 1 1 = = + = + = = = = = = = n X X X X n n n X n X n n i i n i i CH Reilly UCF - IEMS - STA 3032 9 Interval estimation Interval estimates are called confidence intervals (CI). The degree of confidence (or confidence level or confidence coefficient) is given by 1- , the likelihood that the CI will contain the true value of the parameter that is being estimated. Confidence intervals are very commonly used in inferential statistics. CH Reilly UCF - IEMS - STA 3032 10 Interval estimates for We develop interval estimates for because we do not know the true value of this population parameter. (If we did, we would not need an estimate of any kind.) When we do not know the population mean, it is very unlikely that we would know the population variance, 2 ....
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This note was uploaded on 08/22/2010 for the course STA 3032 taught by Professor Sapkota during the Spring '08 term at University of Central Florida.

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JMF-7&8-07 - CH Reilly UCF - IEMS - STA 3032 1...

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