IPS6eCh07_1 - Inference for Distributions for the Mean of a...

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Inference for Distributions for the Mean of a Population IPS Chapter 7.1 © 2009 W.H Freeman and Company
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Objectives (IPS Chapter 7.1) Inference for the mean of a population The t distributions The one-sample t confidence interval The one-sample t test Matched pairs t procedures Robustness Power of the t- test Inference for non-normal distributions
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Sweetening colas Cola manufacturers want to test how much the sweetness of a new 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 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 : > 0 This looks familiar. However, here we do not know the population parameter σ . The population of all cola drinkers is too large. Since this is a new cola recipe, we have no population data. This situation is very common with real data.
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When  σ  is unknown When the sample size is large, the sample is likely to contain elements representative of the whole population. Then s is a good estimate of . Population distribution Small sample Large sample But when the sample size is small, the sample contains only a few individuals. Then s is a mediocre estimate of . The sample standard deviation s provides an estimate of the population standard deviation .
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A 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? Standard deviation  – standard error  s/√n 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 sample results as mean ± SEM. - - = 2 ) ( 1 1 x x n s i SEM = s /√ n <=> s = SEM*√ n s = 8.9*√25 = 44.5
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The  t  distributions Suppose that an SRS of size n is drawn from an N ( µ , σ ) population. When σ is known, the sampling distribution is N ( μ,σ /√ n ). When is estimated from the sample standard deviation s , the sampling distribution follows a t distribution t ( μ , s /√ n ) with degrees of freedom n − 1. is the one-sample t statistic. t = x s n
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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 from s .
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Standardizing the data before using Table D Here, μ is the mean (center) of the sampling distribution, and the standard error of the mean s/√n is its standard deviation (width).
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IPS6eCh07_1 - Inference for Distributions for the Mean of a...

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