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Unformatted text preview: 1 Tests of Hypotheses 2 Statistical hypothesis Statistical hypothesis Statement about a feature of the population (e.g. – the mean) Examples: Mean temperature of healthy adults is 98.6°F (37°c). A certain medication contains a mean of 245 ppm of a particular chemical. Mean number of people that enter a certain restaurant in a day is 125. 3 Example – soft drink bottles A firm that produces a certain soft drink prints on each bottle that it contains 24 oz of drink. It has been suspected that the mean amount per bottle is less than 24 oz . In order to examine this claim, a sample of 100 bottles has been taken and the mean amount per bottle was found to be 23.4 oz. Assume that the standard deviation of the contents of the drink in the bottles is σ=3 oz. Is this an indication that the mean amount of drink in a bottle is less than 24 oz? 4 Set hypotheses: Two types of hypotheses : H the null hypothesis Common beliefs, the claims that are assumed to be true uptodate H Mean contents of soft drink bottle, μ, is 24 oz (μ=24) H 1 the alternative hypothesis Alternative claims that come to challenge the common beliefs 5 Use the sample results to test the hypotheses 6 If is (i) small enough (ii) large enough we will reject H . X H : μ=24 H 1 : μ<24 Sample results: 4 . 23 = X 7 How likely are we to observe such result from a population with mean μ=24? μ=24 oz The distribution of X 4 . 23 = X ? ) 4 . 23 ( = ≤ X p = = = 3 . 100 3 , 24 ~ then 24, If μ μ N X 8 ? ) 4 . 23 ( = ≤ X p = = 3 . 100 3 , 24 ~ that recall μ N X 0228 . ) 2 ( 3 . 24 4 . 23 ) 4 . 23 ( = ≤ = =  ≤ = ≤ Z p Z p X p We are not very likely to observe such result from a population with μ=24 (prob=0.028) 9 Test statistic Z = 2 is an example of a test statistic It measures the distance of the sample results from what is expected if H is true. n X Z σ μ = 10 μ=24 oz 2 4 . 23 = = Z or X 0.0228 Is the value Z=2 unusual under H ? There is only 0.0228 chance of getting values smaller than Z=2 if H is true 11 Pvalue The probability of getting an outcome as extreme or more extreme than the observed outcome. “Extreme” – far from what we would expect if H were true. The smaller the pvalue, the stronger the evidence against H . 12 Level of significance α – significance level. It is the chance we are ready to take for rejecting H while in fact H is true if pvalue≤ α, we say that we reject H at the α significance level. Typically, α is taken to be 0.05 or 0.01 13 In the bottles example: If we required a significance level of α=0.05 then we would reject H pvalue=0.0228<0.05 However, if we required a significance level of α=0.01 then we would not reject H X μ=24 2 4 . 23 = = Z or X 0.028 14 Example – sales of coffee Weekly sales of regular ground coffee at a supermarket have in the recent past varied according to a normal distribution with mean μ=354 units per week and standard deviation...
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 Spring '09
 N/A
 Statistics, Statistical hypothesis testing, H0

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