10_2sam_hypotest

10_2sam_hypotest - Two-sample hypothesis testing, I 9.07...

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Two-sample hypothesis testing, I 9.07 3/09/2004
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• But first, from last time…
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More on the tradeoff between Type I and Type II errors • The null and the alternative: µ ο µ a Sampling distribution of the mean, m, given mean µ a . (Alternative) This is the mean for the systematic effect. Often we don’t know this. Sampling distribution of the mean, m, given mean µ o . (Null)
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More on the tradeoff between Type I and Type II errors • We set a criterion for deciding an effect is significant, e.g. α =0.05, one-tailed. µ ο µ a criterion α =0.05
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More on the tradeoff between Type I and Type II errors Note that α is the probability of saying there’s a systematic effect, when the results are actually just due to chance. = prob. of a Type I error. µ ο µ a criterion α =0.05
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More on the tradeoff between Type I and Type II errors •W h e r e a s β is the probability of saying the results are due to chance, when actually there’s a systematic effect as shown. = prob. of a Type II error. µ
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More on the tradeoff between Type I and Type II errors • Another relevant quantity: 1- β . This is the probability of correctly rejecting the null hypothesis (a hit). µ ο µ a criterion 1−β β
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For a two-tailed test Accept H 0 Reject H 0 Reject H 0 1−β ( correct rejection) α (Type I error) β (Type II error)
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Type I and Type II errors • Hypothesis testing as usually done is minimizing α , the probability of a Type I error (false alarm). • This is, in part, because we don’t know enough to maximize 1- β (hits). • However, 1- β is an important quantity. It’s known as the power of a test. 1−β = P(rejecting H 0 | H a true)
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Statistical power • The probability that a significance test at fixed level α will reject the null hypothesis when the alternative hypothesis is true. = 1 - β • In other words, power describes the ability of a statistical test to show that an effect exists (i.e. that H o is false) when there really is an effect (i.e. when H a is true). • A test with weak power might not be able to reject H o even when H a is true.
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Example: why we care about power • Suppose that factories that discharge chemicals into the water are required to prove that the discharge is not affecting downstream wildlife. • Null hypothesis: no effect on wildlife • The factories can continue to pollute as they are, so long as the null hypothesis is not rejected at the 0.05 level. Cartoon guide to statistics
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Example: why we care about power • A polluter, suspecting he was at risk of violating EPA standards, could devise a very weak and ineffective test of the effect on wildlife. • Cartoon guide extreme example: “interview the ducks and see if any of them feel they are negatively impacted.” Cartoon guide to statistics
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Example: why we care about power • Just like taking the battery out of the smoke alarm, this test has little chance of setting off an alarm. • Because of this issue, environmental regulators
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10_2sam_hypotest - Two-sample hypothesis testing, I 9.07...

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