PSLS.PPT.Ch17 - (unknown PSLS chapter 17 2009 W.H Freeman...

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    Inference for a population mean ( σ  unknown) PSLS chapter 17 © 2009 W.H. Freeman and Company
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Objectives (PSLS chapter 17) Inference for the mean of one population (σ unknown) When σ is unknown The t distributions The t test Confidence intervals Matched pairs t procedures Robustness
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Sweetening colas How is the sweetness of a cola drink is affected by storage? The sweetness loss due to storage was evaluated by 10 professional tasters (by comparing the sweetness before and after storage): Taster Sweetness loss (positive=loss) 1 2.0 2 0.4 3 0.7 4 2.0 5 −0.4 6 2.2 7 −1.3 8 1.2 9 1.1 10 2.3 Obviously, we want to test if storage results in a loss of sweetness, thus H 0 : μ = 0 versus H a : μ > 0 This looks familiar, except that we do not know the population parameter σ . This situation is very common with real data.
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When  σ  is unknown When the sample size is very large, s is a very good estimate of σ . Population distribution Small sample Large sample When the sample size is very small, s is a more mediocre estimate of σ . The sample standard deviation s provides an estimate of the population standard deviation σ .
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The  t  distributions We test a null and alternative hypotheses with 1 random sample of size n from a Normal population N ( µ , σ ): When σ is known, the sampling distribution is Normal N ( μ , σ /√ n ). When σ is estimated from the sample standard deviation s , then the sampling distribution follows a t distribution t ( μ , s /√ n ) with degrees of freedom n − 1.
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When n is very large, s is a very good estimate of σ and the corresponding t distributions are very close to the Normal distribution. The t distributions become wider for smaller sample sizes, reflecting the lack of precision in estimating σ from s .
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A medical study examined the effect of a new medication on the seated systolic blood pressure. The results, presented as mean ± SEM for 25 patients, are 113.5 ± 8.9. What is the standard deviation s of the sample data? For a sample of size n , the sample standard deviation s is: n − 1 is the “degrees of freedom.” The value s /√ n is called the standard error of the mean SEM . Scientists often present their sample results as the mean ± SEM.
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