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Unformatted text preview: Kata Bognar firstname.lastname@example.org Economics 41 Statistics for Economists UCLA Spring 2010 Midterm 2 - Version A- answer key - Part I - Multiple Choice Questions (3 points each) 1. Among the contestants in a competition there are 5 women and 5 men. Four winners are selected by a jury. The first winner selected is a male. What is the probability that there is at least one female among the remaining three winners? C (a) 29/30 (b) 79/84 (c) 20/21 (d) none of the above Explanation : After one male is already picked, the three remaining winners are selected from 4 men and 5 women. The probability that there is at least one female among the three winners is equal to 1 minus the probability that there is none. To calculate the probability that there is no female among the last three winners, you have to count (i) the number of ways in which any 3 winner can be selected, (ii) the number of ways in which 3 men can be selected and finally divide the number in (ii) by the number in (i). There are ( 9 3 ) = 84 different ways in which the 3 winners can be selected and there are ( 4 3 ) = 4 different ways in which 3 men can be selected. Thus, the probability that no woman is selected is exactly 4 84 = 1 21 and the probability that at least one woman is selected is 1- 1 / 21 = 20 / 21 . 2. Which of the following is an example of a discrete random variable? C (a) The weight of a randomly selected box of cookies (b) The length of a randomly selected window frame (c) The number of horses owned by a randomly selected farmer (d) The distance from home to work for a randomly selected worker Explanation: The variables in (a), (b) and (d) are continuous. 3. Sixty percent of children in a school do not have cavities. Let X be the number of children in a random sample of 20 children selected from this school who do not have cavities. The mean of the probability distribution of X is: D (a) 8 (b) 18 (c) 10 (d) none of the above Explanation: The random variable X follows a binomial distribution with parameters n = 20 and p = 0 . 6 . The mean of a binomial random variable is x = np = 20 . 6 = 12 . 1 4. There are N balls in an urn, they are either white or red. A sample of n balls is selected without replacement. Denote by X the number of red balls in the sample. D (a) X has a binomial distribution, regardless of the values of n and N. (b) X has a bernoulli distribution, regardless of the values of n and N. (c) X has a normal distribution, regardless of the values of n and N. (d) none of the above Explanation: The sampling is without replacement so X does not follow a binomial distribution. For n = 1, X has a bernoulli distribution but for other values of n that is not true. Finally, X is discrete so (c) cannot be the answer either....
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