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Chapter18

The Basic Practice of Statistics (Paper) & Student CD

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Inference for a population mean BPS chapter 18 © 2006 W.H. Freeman and Company

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Objectives (BPS chapter 18) Inference about a Population Mean Conditions for inference The t distribution The one-sample t confidence interval Using technology Matched pairs t procedures Robustness of t procedures
Conditions for inference about a mean We can regard our data as a simple random sample (SRS) from the population. This condition is very important. Observations from the population have a Normal distribution with mean μ and standard deviation σ . In practice, it is enough that the distribution be symmetric and single-peaked unless the sample is very small. Both and standard deviation are unknown.

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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.
When σ is unknown When the sample size is very large, the sample is likely to contain elements representative of the whole population. Then s is a very 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 more mediocre estimate of . The sample standard deviation s provides an estimate of the population standard deviation .

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Example: 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? Standard deviation s standard error of the mean 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 their sample results as the mean ± SEM. ! " " = 2 ) ( 1 1 x x n s i SEM = s / n <=> s = SEM* n s = 8.9* 25 = 44.5
The t distributions We test a null and alternative hypotheses with one sample of size n from a normal population N ( µ , σ ): When σ is known, the sampling distribution is normal N ( μ , / n ). When is estimated from the sample standard deviation s , then the sampling distribution follows a t distribution t ( , s / n ) with degrees of freedom n 1.

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Chapter18 - Inference for a population mean BPS chapter 18...

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