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Unformatted text preview: Chapter 7 SAMPLING DISTRIBUTIONS The idea of sampling distributions is the key concept in (frequentist) statistical inference. The idea allows us to use probability models to make inferential statements from the data. This chapter can be viewed as the transition from the theory of probability to the theory of statistics. The central limit theorem explains the ubiquity of the normal distribution (“bell-shaped curve”) in many natural phenomena: e.g. distribution of weights, lengths, IQ’s, etc. 7.1 Statistics and Estimators 7.2 Linear Combinations 7.3 The Central Limit Theorem 7.4 The Law of Large Numbers 7.5 Normal Approximation of the Binomial Distribution 7.1 STATISTICS AND ESTIMATORS Example (1) A test of 90 automobile batteries shows that 75 are still in service after 6 months of use. If conditions remain the same, in the long run, what proportion of batteries will last 6 months or more? Model: Assume Bernoulli trials. X is binomial B(90, p) for an unknown p . ˆ p = 75 90 Question: How far is ˆ p from p ? 1 Example(2) The life times of 2 types of batteries are measured. The results were: ¯ X = 6 . 5 months with n X = 10 and S X = 2 ¯ Y = 8 months with n Y = 12 and S Y = 2 . 6 Can we believe that type 2 is really longer-lasting than type 1? Model: X 1 ,X 2 ,...,X 10 are indep N ( μ X ,σ 2 ) Y 1 ,Y 2 ,....,Y 12 are indep N ( μ Y ,σ 2 ) Question: How different can ( ¯ X- ¯ Y ) be from ( μ X- μ Y )? In these two examples ˆ p and ¯ X- ¯ Y are “statistics” i.e. they are computed from the data. They are also estimators . Estimators are statistics that yield estimates of un- known parameters or functions thereof, here p and μ X- μ Y , respectively. Other statistics/estimators are: ˜ X sample median — estimates F- 1 ( 1 2 ) S 2 sample variance σ 2 S sample std. dev. σ Q 1 sample lst quartile F- 1 ( 1 4 ) Also ¯ X 2 S 2 is an estimate of α S 2 ¯ X β in a gamma model — see Notes Sec 6.2....
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This note was uploaded on 06/12/2008 for the course ENGRD 2700 taught by Professor Staff during the Spring '05 term at Cornell.
- Spring '05