What’s in the Bag?How Strong is the Evidence?The p-valueTwo BagsSlide 5Slide 6Slide 7A Two-Sided TestSlide 9Slide 10Slide 11Slide 12Slide 13Slide 14ShortcutSlide 16Two Explanations of Unusual ObservationsSlide 18Slide 19Slide 20Slide 21Case Study 3What’s in the Bag?Lecture 5Section 1.4.3Wed, Aug 29, 2007How Strong is the Evidence?Rather than give an accept/reject answer, we may ask a different question:How strong is the evidence against H0?We use the p-value to measure this.The p-valueIn the Bag A/Bag B example, if the selected token is worth $50, what is the p-value?Two Bags-1000 10 20 30 40 60 100050Bag A-1000 10 20 30 40 60 100050Bag BObservedvalueTwo Bags-1000 10 20 30 40 60 100050Bag A-1000 10 20 30 40 60 100050Bag BAt least as extreme as 50Two Bags-1000 10 20 30 40 60 100050Bag A-1000 10 20 30 40 60 100050Bag Bp-value = 2/20 = 0.10At least as extreme as 50The p-valueIf the selected token is worth $30, what is the p-value?Keep in mind, we may always compute the p-value regardless of our decision about which hypothesis to accept.A Two-Sided TestBag F1 2 3 4 65Bag E8 10971 2 3 4 658 1097A Two-Sided TestIf the selected token is worth $8, what is the p-value?First, what is the direction of extreme?Which values are at least as extreme as 8?A Two-Sided TestBag F1 2 3 4 65Bag E8 10971 2 3 4 658 1097ObservedvalueA Two-Sided TestBag F1 2 3 4 65Bag E8 10971 2 3 4 658 1097ObservedvalueEquallyextremeA Two-Sided Test1 2 3 4 65Bag EBag F8 10971 2 3 4 658 1097At least as extreme as 8At least as extreme as 8A Two-Sided Test1 2 3 4 65Bag EBag F8 10971 2 3 4 658 1097p-value = 12/30 = 0.40The p-valueIf the selected token is worth $1, what is the p-value?ShortcutIn a two-sided test, if the null distribution is symmetric, then you can compute the probability in one direction, and then double it to get the p-value.The p-valueA small p-value is strong evidence against the null hypothesis.Why?Two Explanations of Unusual ObservationsThe null hypothesis leads us to a certain expectation of what the data will show.If the data deviate from our expectation, then we need to explain that deviation.Two Explanations of Unusual ObservationsTwo explanations:The null hypothesis is true; the deviation is due to chance.The null hypothesis is false; we had the wrong expectation in the first place.Two Explanations of Unusual ObservationsChance is a fine explanation if the deviation is small.Chance is not a very good explanation if the deviation is large.That’s where the p-value comes in.Two Explanations of Unusual ObservationsSmall deviations are consistent with the null hypothesis.They have high probability (large p-values) if the null hypothesis is true.Two Explanations of Unusual ObservationsLarge deviations are inconsistent with the null hypothesis.They have low probability (small p-values) if the null hypothesis is true.Case Study 3Long-term Proton Pump Inhibitor Therapy and Risk of Hip
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