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PDB_Stat_100_Lecture_22_Printable

# PDB_Stat_100_Lecture_22_Printable - STA 100 Lecture 22 Paul...

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Unformatted text preview: STA 100 Lecture 22 Paul Baines Department of Statistics University of California, Davis February 28th, 2011 Admin for the Day I Homework 6 posted, due Wednesday at 5pm! I Project Proposals due Today! (email or paper, either is fine) I Three possible project datasets are posted References for Friday: Rosner, Ch 10.1-10.6 (7th Ed.) References for Wednesday: Rosner, Ch 11 (7th Ed.) Example Example: From the midterm, there were two sloth populations (living in two separate regions). Previously, I just told you that the second population had a probability p = 0 . 06 of injury. In reality, we don’t know this. Let p 1 be the probability of injury in region 1, and p 2 be the probability of injury in region 2. We want to ask: is p 1 = p 2 ? i.e., do the two regions pose the same risks to the resident sloths? Example So, if we let X i be the number of injured sloths in region i then: I X 1 ∼ Bin ( n 1 , p 1 ), I X 2 ∼ Bin ( n 2 , p 2 ). (Note: the regions may different numbers of sloths so n 1 doesn’t need to be the same as n 2 ). We want to test: I H : p 1 = p 2 , vs., I H 1 : p 1 6 = p 2 . Two-Sample Binomial Testing Recall that for Binomial and Poisson one-sample testing we had two methods: (i) approximate (using the t- distribution), and, (ii) exact (using a fancier method that R did for us). The same is true for two-sample testing for Binomial and Poisson. Last time we looked at approximate methods (which you can understand and implement yourself), Today we look at an exact method (the details of which you won’t need to know). First, we recap the approximate methods. . . Two-Sample Binomial Test 1. If p is the same for both populations, we compute the ‘combined’ estimate for p , using both groups: ˆ p = x 1 + x 2 n 1 + n 2 2. We use the test statistic: z = ˆ p 1- ˆ p 2 r ˆ p (1- ˆ p ) 1 n 1 + 1 n 2 3. Under H : p 1 = p 2 , the test statistic is (approx.) ∼ N (0 , 1). 4. We compute the p- value (one- or two-sided) as usual: P ( Z > | z | ) (two-sided), or, P ( Z < z ) (one-sided less-than), or, P ( Z > z ) (one-sided greater-than). 5. Decide whether to accept or reject H (is p- value less than α ?) 6. Interpret your decision for the specific problem. Continuity Corrections Remember that this test requires an approximation. It turns out that we can obtain a better approximation with a very minor modification known as a continuity correction . (again: we have an exact test so this is not usually necessary – but you often see it in academic papers, so it is worth knowing. See Rosner Ch 10.2 for more details if interested). Binomial Test w/ Continuity Correction 1. If p is the same for both populations, we compute the ‘combined’ estimate for p , using both groups: ˆ p = x 1 + x 2 n 1 + n 2 2. We use the test statistic: z = (ˆ p 1- ˆ p 2 )- sign (ˆ p 1- ˆ p 2 ) × 1 2 n 1 + 1 2 n 2 r ˆ p (1- ˆ p ) 1 n 1 + 1 n 2 where: sign (ˆ p 1- ˆ p 2 ) = 1 if ˆ p 1- ˆ p 2 >- 1 if ˆ p 1- ˆ p 2 < 3. Under H , the test statistic is (approx.), the test statistic is (approx....
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PDB_Stat_100_Lecture_22_Printable - STA 100 Lecture 22 Paul...

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