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notes515fall10chap8 - STAT 515 Chapter 8 Hypothesis Tests...

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STAT 515 -- Chapter 8: Hypothesis Tests • CIs are possibly the most useful forms of inference because they give a range of “reasonable” values for a parameter. • But sometimes we want to know whether one particular value for a parameter is “reasonable.” • In this case, a popular form of inference is the hypothesis test . We use data to test a claim (about a parameter) called the null hypothesis . Example 1: We claim the proportion of USC students who travel home for Christmas is 0.95. Example 2: We claim the mean nightly hotel price for hotels in SC is no more than \$65. • Null hypothesis (denoted H 0 ) often represents “status quo”, “previous belief” or “no effect”. • Alternative hypothesis (denoted H a ) is usually what we seek evidence for. We will reject H 0 and conclude H a if the data provide convincing evidence that H a is true. Evidence in the data is measured by a test statistic .

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A test statistic measures how far away the corresponding sample statistic is from the parameter value(s) specified by H 0 . If the sample statistic is extremely far from the value(s) in H 0 , we say the test statistic falls in the “rejection region” and we reject H 0 in favor of H a . Example 2: We assumed the mean nightly hotel price in SC is no more than \$65, but we seek evidence that the mean price is actually greater than \$65. We randomly sample 64 hotels and calculate the sample mean price X . Let n X Z / 65 σ = be our “test statistic” here. Note: If this Z value is much bigger than zero, then we have evidence against H 0 : μ 65 and in favor of H a : μ > 65. Suppose we’ll reject H 0 if Z > 1.645. If μ really is 65, then Z has a standard normal distribution. (Why?) Picture:
If we reject H 0 whenever Z > 1.645, what is the probability we reject H 0 when H 0 really is true ? P( Z > 1.645 | μ = 65) = This is the probability of making a Type I error (rejecting H 0 when it is actually true). P(Type I error) = “level of significance” of the test (denoted α ). We don’t want to make a Type I error very often, so we

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notes515fall10chap8 - STAT 515 Chapter 8 Hypothesis Tests...

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