9sngl_sam_hypts2

9sngl_sam_hypts2 - Single sample hypothesis testing, II...

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Single sample hypothesis testing, II 9.07 3/02/2004
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Outline • Very brief review • One-tailed vs. two-tailed tests • Small sample testing • Significance & multiple tests II: Data snooping • What do our results mean? • Decision theory and power
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Brief review • Null and alternative hypothesis – Null: only chance effects – Alternative: systematic + chance effects • Assume the null is true • Given this assumption, how likely is it that we’d see values at least as extreme as the ones we got? • If it’s highly unlikely, reject the null hypothesis, and say the results are statistically significant. – The results are due to a combination of chance and a systematic effect.
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Key Concepts •H 0 and H a are contradictory (mutually exclusive) • Support for H a can only be obtained indirectly -- by rejecting H 0 • Rationale: – We can never prove anything true, but we can prove something false – We know the value of the parameter given H 0 but not given H a
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Why bother with H a at all? • The alternative hypothesis describes the condition that is contrary to the null hypothesis, and this can be directional or non-directional – Directional: The effect only occurs in a specific direction -- increases or decreases – Non-directional: The effect may be greater or less than a population parameter
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Outline • Very brief review • One-tailed vs. two-tailed tests • Small sample testing • Significance & multiple tests II: Data snooping • What do our results mean? • Decision theory and power
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A Tale of Two Tails • Directional hypotheses are called one-tailed – We are only interested in deviations at one tail of the distribution • Non-directional hypotheses are called two- tailed – We are interested in any significant deviations from H 0
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The p-value for a test of H o : = o against: H a : μ> μ o is prob H a : μ< μ o is prob H a : μ μ o is prob z |z| z Figure by MIT OCW. μ μ
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How do you decide to use a one- or two-tailed approach? p z obt 2p A one-tailed approach is more liberal -- it is more likely to declare a result significant. –t crit = 1.69 5%, one-tailed –t crit = 2.03 5%, two-tailed There’s no one right answer as to which test to use. People will debate this point.
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The moderate approach: • If there’s a strong, prior, theoretical expectation that the effect will be in a particular direction (A>B), then you may use a one-tailed approach. Otherwise, use a two-tailed test. • Because only an A>B result is interesting, concentrate your attention on whether there is evidence for a difference in that direction. – E.G. does this new educational reform improve students’ test scores? – Does this drug reduce depression?
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9sngl_sam_hypts2 - Single sample hypothesis testing, II...

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