# 7.1 - Inferencefordistributions(7.1)...

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Inference for distributions (7.1) for the mean of a population with unknown  variance

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Inference for the mean of one population When σ is unknown The t distributions t confidence interval t test Matched pairs t procedures Robustness Inference for non-normal distributions
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 We want to test if storage results in a loss of sweetness, thus: H 0 : μ = 0 vs. H a : > 0 μ is the mean loss This looks familiar. However, here we do not know the population parameter σ . 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 known, a 95% confidence interval for µ is When σ is unknown, it needs to be estimated.
When  σ   is unknown The sample standard deviation s provides an estimate of the population standard deviation . For a sample of size n , the sample standard deviation s is: n − 1 is the “degrees of freedom.” The value is called the standard error of the mean.

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When the sample size is large, s is a good estimate of σ . But when the sample size is small, s is a more mediocre estimate of .

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For large enough sample size, , and so is an approximate 95% confidence interval for when the sample is sufficiently large.
is an exact 95% confidence interval for since has standard normal distribution. To derive an exact confidence interval for in the case where is unknown, we need to know the probability distribution of and replace 1.96 with the 97.5th percentile of that distribution.

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The  t  distributions Suppose that an SRS of size n is drawn from an N ( µ , σ ) population. Then follows a t distribution with n − 1 degrees of freedom. This statistic is called a one-sample t statistic. t = x s n
The t distributions are more spread out for smaller degrees of freedom, reflecting the lack of precision in estimating σ from s . When n is very large, s is a very good estimate of and the corresponding t distributions are very close to the standard normal distribution.

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An exact level C confidence interval forμ when σ is unknown is is the value such that P(T > ) = (1 - C)/2, where T has a t distribution with n-1 degrees of freedom.
Table D Table D shows the z -values and t -values for some selected upper tail probabilities.

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## This note was uploaded on 09/04/2010 for the course STAT 131 taught by Professor Isber during the Spring '08 term at University of California, Berkeley.

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7.1 - Inferencefordistributions(7.1)...

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