AK5_230 - ECO 230 Marianna Kudlyak Economic Statistics...

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Unformatted text preview: ECO 230 Marianna Kudlyak Economic Statistics Spring 2007 Answer Key to Homework #5 Total points: 10. Graded exercises: Ex.3 (3 points), Ex.6 (3 points), Ex. 4 (2.5 points), and 0.5 points were added for the presence of each of the remaining 3 exercises. Exercise 1. Newbold, Carlson and Thorne, Problem 5.67 p.165. Exercise 2. Newbold, Carlson and Thorne, Problem 5.69 p.165. Exercise 3. a) If 40% of the balls in a certain box are red and if 5 balls are selected from the box at random with replacement, what is the probability that more than 3 red balls will be selected? Let X be a random variable that denotes a number of red balls among 5 selected balls. The distribution of X is binomial with probability of success 0.4 (since the sampling is with replacement). P(X>3) = P(X=4) + P(X=5), where P(X=x) = C 5 x 0.4 x (1-0.4) 5-x b) If 4 of the balls in a certain box are red and 7 are black and if 5 balls are selected from the box at random without replacement, what is the probability that more than 3 red balls will be selected? Here we should assume that there are no other balls except 4 red and 7 black balls in the box. Let X be a random variable that denotes a number of red box among the selected 5 balls. Since the sampling is without replacement, the distribution of X is hypergeometric. P(X>3)=P(X=4) + P(X=5), P(X=5) = 0 since at most 4 red balls can be selected. Then 7 4 5 7 1 4 4 ) 4 ( ) 3 ( + = = = > C C C x P X P c) A company receives a shipment of 16 items. A random shipment of 4 items is selected, and the shipment is rejected if any of these items proves to be defective. What is the probability of accepting the shipment containing 1 defective item? Let X be a random variable that denotes a number of defective items among the selected 4. The distribution of X is hypergeometric. P(‘shipment is accepted’) = 1 – P(‘shipment is rejected’) = 1 – P(X>0) = 1- {1 - P(X=0)} = P(X=0) = (C 1 ) (C 16-1 4 )/( C 16 4 ) Exercise 4. Consider the following game. You pay $1 to roll three dice and choose a number. If your number is rolled on any of the dice, you get your dollar back plus one dollar for each of the times your number came up. For instance, if your number comes up on two of the three dice, than you get a total of three dollars back. On average, how much would you expect to win at than you get a total of three dollars back....
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AK5_230 - ECO 230 Marianna Kudlyak Economic Statistics...

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