What s in the Bag Lecture 5 Section 1 4 3 Wed Aug 29 2007 How 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 value In the Bag A Bag B example if the selected token is worth 50 what is the pvalue Two Bags Bag A 1000 10 20 30 40 50 60 1000 Observed value Bag B 1000 10 20 30 40 50 60 1000 Two Bags Bag A At least as extreme as 50 1000 10 20 30 40 50 60 1000 1000 10 20 30 40 50 60 1000 Bag B Two Bags p value 2 20 0 10 Bag A At least as extreme as 50 1000 10 20 30 40 50 60 1000 1000 10 20 30 40 50 60 1000 Bag B The p value If 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 Test Bag E 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Bag F A Two Sided Test If 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 Test Bag E 1 2 3 4 5 6 7 8 9 10 Observed value Bag F 1 2 3 4 5 6 7 8 9 10 A Two Sided Test Bag E 1 2 3 4 5 6 7 8 9 10 Equally extreme Observed value Bag F 1 2 3 4 5 6 7 8 9 10 A Two Sided Test At least as extreme as 8 At least as extreme as 8 Bag E 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Bag F A Two Sided Test p value 12 30 0 40 Bag E 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Bag F The p value If the selected token is worth 1 what is the p value Shortcut In 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 value A small p value is strong evidence against the null hypothesis Why Two Explanations of Unusual Observations The 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 Observations Two 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 Observations Chance 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 Observations Small deviations are consistent with the null hypothesis They have high probability large p values if the null hypothesis is true Two Explanations of Unusual Observations Large deviations are inconsistent with the null hypothesis They have low probability small p values if the null hypothesis is true Case Study 3 Long term Proton Pump Inhibitor Therapy a nd Risk of Hip Fracture
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