Lecture 4 Section 1.4.3 Wed, Jan 19, 2005 What’s in the Bag? Lecture 4 Section 1.4.3 Wed, Jan 19, 2005

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?

The p-value p-value – The likelihood of getting by chance, if H0 is true, a value at least as extreme as the one observed.

Two Bags If the selected token is worth $50, what is the p-value?

Two Bags Bag A -1000 10 20 30 40 50 60 1000 Bag B -1000 10 20 30 40 50 60 1000

Two Bags Bag A -1000 10 20 30 40 50 60 1000 Bag B -1000 10 20 30 40 50 60 1000

Two Bags p-value = 2/20 = 0.10 Bag A Bag B -1000 10 20 30 40 50 60

Two More Bags Bag E 1 2 3 4 5 6 7 8 9 10 Bag F 1 2 3 4 5 6 7 8 9 10

Two More Bags If the selected token is worth $8, what is the p-value?

Two More Bags Bag E 1 2 3 4 5 6 7 8 9 10 Bag F 1 2 3 4 5 6 7 8 9 10

Two More Bags p-value = 12/30 = 0.40 Bag E Bag F 1 2 3 4 5 6 7 8 9 10

The p-value If the null hypothesis is true, then Therefore, The observed data is likely to be close to what the null hypothesis predicts. Therefore, Large discrepancies are less likely than small discrepancies. The p-value measures the likelihood of a discrepancy at least as large as the one observed.

Interpretation of the p-value Smaller p-values Larger p-values 1 p-values

Interpretation of the p-value More extreme values Less extreme values 1 p-values

Interpretation of the p-value Larger discrepancy Smaller discrepancy 1 p-values

Interpretation of the p-value Stronger evidence against H0 Weaker evidence against H0 1 p-values

Interpretation of the p-value Statistically more significant Statistically less significant 1 p-values

Two Explanations For any discrepancy between the evidence and what is predicted by the null hypothesis, there are always two explanations: The discrepancy occurred by chance (random sampling) even though the null hypothesis is true. The discrepancy occurred because the null hypothesis is false. Given the evidence, which explanation is more believable?

Two Explanations The null hypothesis gets the benefit of the doubt. Therefore, If the discrepancy is small (p-value is large), then we go with the first explanation and accept H0. If the discrepancy is large (p-value is small), then we go with the second explanation and reject H0.

Let’s Do It! Let’s do it! 1.9, p. 32 – p-value for a One-Sided Rejection Region to the Left. Let’s do it! 1.10, p. 34 – p-value for a Two-Sided Rejection Region. Example 1.9, p. 36 – Can You Picture the p-value?

Let’s Do It! Let’s do it! 1.7, p. 26 – Chromium Supplements. Let’s do it! 1.8, p. 27 – Three Studies. Let’s do it! 1.11, p. 39 – Machine A or Machine B?