spring2008stat201exam2-version1KEY

spring2008stat201exam2-version1KEY - Stat 201 Exam 2 Spring...

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201 Exam 2 Spring 2008 Version 1 KEY Page 1 of 9 1. In the U.S., 12% of the population is black. Suppose, of all U.S. families that have a brother and a sister, you randomly select one such family. Let X represent the event ‘the brother is black’. Let Y represent the event ‘the sister is black.’ [Note that some U.S. families have a child who is adopted (but most don’t).] Remark: This exercise was anticipated by exercises 1 and 2 on the practice exam; it is just like scenario 3 of lab 6 and exercise 1 of lecture 12; the ideas were covered in various exercises of homework 6, and throughout lectures 10, 11, and 12. (a) Are the two events disjoint? Circle the best answer: [3 pts] (i) Yes, because both siblings could be black. (ii) No, because both siblings could be black. (iii) Yes, because each sibling is selected from the same family. (iv) No, because each sibling is selected from the same family. (v) Yes, because there’s a 12% chance the parents are black. (vi) No, because there’s a 12% chance the parents are black. It is certainly possible for the brother to be black, and for the sister to also be black. In fact, since about 12% of families are black, there’s about a 12% chance of finding a black brother along with a black sister. Symbolically, P(X and Y) = about 12%. Since P(X and Y) 0, the events aren’t disjoint. (b) Are the two events independent? Circle the best answer: [3 pts] (i) Yes, because both siblings could be black. (ii) No, because both siblings could be black. (iii) Yes, because each sibling is selected from the same family. (iv) No, because each sibling is selected from the same family. (v) Yes, because there’s a 12% chance the parents are black. (vi) No, because there’s a 12% chance the parents are black. If the first sibling is black, then most likely you’ve selected a black family, in which case there would be a nearly 100% chance the other sibling is black; whereas, if the first sibling is not black, then you’ve most likely selected a non-black family, in which case there is a nearly zero chance the other sibling is black. So, knowing the outcome of X changes your estimated likelihood of Y. Symbolically, P(Y) = .12, but P(Y given X) = nearly 100% (c) Suppose that you first look at the brother, but not yet at the sister. If you find that the brother is black, then, given that fact, which of the following is most reasonably the probability that the sister will then be black? [3 pts] (i) zero (iv) . 144 (ii) . 12 (v) . 096 (iii) . 24 (vi) close to 100% If the first sibling is black, you have most likely selected a black family. The only reason the other sibling might not also be black is if one of the siblings was adopted (or if perhaps the mother had the siblings with different fathers of different races), either of which is a very unlikely scenario which only happen
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This note was uploaded on 02/01/2009 for the course STAT 36-201 taught by Professor Gordon during the Fall '08 term at Carnegie Mellon.

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spring2008stat201exam2-version1KEY - Stat 201 Exam 2 Spring...

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