This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: PSTAT 120C: Solutions to Assignment # 5 May 23, 2006 1. The distribution of the committee compositions is multinomial with probabilities 0.4, 0.4, and 0.2. (a) The probability is P { X 1 = 2 ,X 2 = 2 ,X 3 = 1 } = 5 2 2 1 ( . 4) 2 ( . 4) 2 ( . 2) = 30( . 4) 4 ( . 2) = . 1536 (b) This event is the union of the disjoint events, { 4, 5, or 6 Independents } or { 3 Independents and 2, 1, or 0 Democrats } or { 2 Independents and 1 or 0 Democrats } or { 1 Independent and no Democrats } . One way to do this calculation is to condition on the number of Independents. First, the number of Independents is marginally binomial so P { I = k } = 6 k . 2 k . 8 6 k Independents 6 5 4 3 2 1 Probability 6 . 4 10 5 0.0015 0.015 0.082 0.246 0.393 If there are 3 Independents, then the distribution of the Democrats is Bin(3 ,. 5). ( p = 1 / 2 because the probability of one of the other three being a Republican or Democrat is the same.) Therefore, P { D = 0 , 1 , 2  I = 3 } = 1 8 + 3 8 + 3 8 = 7 8 Likewise, if there are two independents then D Bin(4 , 1 / 2) P { D = 0 , 1  I = 2 } = 1 16 + 1 4 = 5 16 If there is one independent then D Bin(5 , 1 / 2) P { D = 0  I = 1 } = 1 32 Now we need to add all these things together, multiplying by the marginal probabilities, P { I > D } = P { I = 4 , 5 , 6 } + P { I = 3 } P { D = 0 , 1 , 2  I = 3 } + + P { I = 2 } P { D = 0 , 1  I = 2 } + P { I = 1 } P { D = 0  I = 1 } = 0 . 000064 + 0 . 0015 + 0 . 015 + +( . 082)(7 / 8)+ + ( . 246)(5 / 16) + ( . 393)(1 / 32) = . 017 + . 0718 + . 0769 + . 0123 = 0 . 178 1 2. Here is a table with the expected values and 2 statistics Number of Games 4 5 6 7 Count 11 8 12 24 Expected 6.875 13.75 17.1875 17.1875 ( O E ) 2 /E 2.475 2.405 1.566 2.70 The X 2 = 9 . 145. Comparing this with the 2 with three degrees of freedom critical value = 7.81. Therefore, there is a significant deviation from proposed model....
View
Full
Document
 Spring '07
 idk

Click to edit the document details