Lecture08

# Lecture08 - Sweetening colas Cola manufacturers want to...

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Inference for Distributions Inference for the Mean of a Population Section 7.1 © 2009 W.H Freeman and Company 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 ! . " 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 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 more mediocre estimate of . The sample standard deviation s provides an estimate of the population standard deviation . A study examined the effect of a new medication on the seated systolic blood pressure. The results, presented as mean ± SE for 25 patients, are 113.5 ± 8.9. What is the standard deviation s of the sample data? Standard error 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 or SE . Scientists often present sample results as mean ± SE. SE = s / " n <=> s = SE* " 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 ( " , # ) population. ! When ! is known, the sampling distribution is N ( μ , / " n ). ! When is estimated from the sample standard deviation s , the sampling distribution follows a t distribution with degrees of freedom n " 1. is the one-sample t statistic. When n is very large, s is a very good estimate of 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 . The one-sample t- confidence interval The level C confidence interval is an interval with probability C of containing the true population parameter. We have a data set from a population with both and unknown. We use to estimate , and s to estimate , using a t distribution, df = n ! 1. C t * ! t * m m Practical use of t : t * ! C is the area between ! t * and t *. ! We find t * in the line of Table D for df = n ! 1 and confidence level C . ! The margin of error m is: Red wine, in moderation Drinking red wine in moderation may protect against heart attacks. The polyphenols it contains act on blood cholesterol and thus are a likely cause.
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Lecture08 - Sweetening colas Cola manufacturers want to...

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