notes509fall11sec42

notes509fall11sec42 - STAT 509 Sections 4.2-4.3 Hypothesis...

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STAT 509 – Sections 4.2-4.3 – Hypothesis Testing • 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 assume a milk carton filling machine produces cartons with a mean weight of 260 g. • Question: Is this true, or is the process overfilling the cartons on average? • If the engineer finds reason to believe the mean weight is greater than 260, he/she would correct the process. Engineer’s Decision Correct process Leave alone Actual wt = 260 Type I error OK Actual wt > 260 OK Type II error
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Hypotheses and Types of Errors • 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. NOTE: There are two types of wrong decisions in a hypothesis test: (1) Type I error : We reject null hypothesis when H 0 is true. (2) Type II error : We fail to reject the null hypothesis when the alternative hypothesis is true. Statistician’s Decision Truth Reject H 0 Fail to reject H 0 H 0 is true Type I error OK H a is true OK Type II error Let α = P(type I error), β = P(type II error) • The power of the test is then 1-β. Power = Probability of rejecting H 0 when H 0 is false. Idea : We will reject H 0 and conclude H a if the data provide convincing evidence that H a is true.
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Evidence in the data is measured by a test statistic . • 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 : Our claim assumed the mean milk carton weight is no more than 260 g, but we seek evidence that the mean weight is actually greater than 260. We randomly sample 49 cartons and calculate the sample mean weight. Assuming we know σ , let n Y Z / 260 σ - = be our “test statistic” here. Note:
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notes509fall11sec42 - STAT 509 Sections 4.2-4.3 Hypothesis...

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