HW5-soln - IE 111 Fall 2007 Homework #5 Due Wednesday...

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IE 111 Fall 2007 Homework #5 Due Wednesday October 10 in class Question 1 Let X be a Binomial random variable with n=100, p=.7 (a) Which one of the following Excel formulas will calculate Pr(X=100) ? =BINOMDIST(0,100,.7,TRUE) =BINOMDIST(0,100,.7,FALSE) =BINOMDIST(100,100,.7,TRUE) = BINOMDIST(100,100,.7,FALSE) The syntax is BINOMDIST(x,N,p,TRUE) for cumulative prob’s and BINOMDIST(x,N,p,False) for PMF values. Thus the answer is BINOMDIST(100,100,.7,FALSE) (b) Which one of the following Excel formulas will calculate Pr(X>=65) ? =BINOMDIST(65,100,.7,TRUE) =1-BINOMDIST(65,100,.7,TRUE) =BINOMDIST(65,100,.7,FALSE) =1-BINOMDIST(65,100,.7,FALSE) =BINOMDIST(64,100,.7,TRUE) = 1-BINOMDIST(64,100,.7,TRUE) =BINOMDIST(64,100,.7,FALSE) =1-BINOMDIST(64,100,.7,FALSE) Question 2 The faster we force our drilling machine to go, the more likely it is to break. Let's suppose that each day is an independent trial, and we can set the probability of the machine breaking to any value “p” that we choose (by controlling the drill speed). Suppose we want Pr(time until next breakdown > 30 days) = 90%. What value of “p” will just satisfy this requirement? P(X>30) = P(No breakdowns in first 30 days) = (1-p) 30 Set (1-p) 30 = 0.9 and solve: (1-p)= (0.9) 1/30 = 0.996494 so p= 0.003606 Question 3 For each description, write the name of the appropriate distribution. Some distributions may be used more than once; others might not be used at all. (a) The number of Democrats, if we randomly select a committee of 10 US Senators out of the set of 100 Senators. (b) The number of bad resistors in a roll of 1000 resistors. (c) The result of a single die roll. (d) The number of children that a couple has until they have a girl (ignore twins). (e) The number of people that show up for an airline flight. (f) The number of free throws I must shoot until I make a total of ten. (g) A Negative Binomial (also called Pascal) distribution with r=1.
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(a) Hypergeometric (b) Binomial (c) Discrete uniform (d) Geometric (e) Binomial (f) Negative Binomial (or Pascal) (g) Geometric Question 4 Clarinet reeds from Company A come in boxes of 8, and (historically) 45% of their reeds are good, on average. Clarinet reeds from Company B come in boxes of 12, and historically they have 35% good reeds. The two companies charge the same amount per box. We can assume all reeds are independent. If you were playing a single concert (for which you need two good reeds) and could only bring one box of reeds, which company gives you a better chance of having at least two good reeds? N= 8 12 p= 0.45 0.35 P(2 or more good) 0.936819 0.957559 Uses formula of the form 1-BINOMDIST(1,N,P,TRUE) OR Company A = 1- 8 C 0 (0.45) 0 (1-0.45) 8-0 - 8 C 1 (0.45) 1 (1-0.45) 8-1 Company B = 1- 12 C 0 (0.35) 0 (1-0.35) 12-0 - 12 C 1 (0.35) 1 (1-0.35) 12-1 Question 5 A study to evaluate a new medicine has been tracking a set of people for the last few years. As of last year’s survey, 50 people remained in the study. It’s time for another
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HW5-soln - IE 111 Fall 2007 Homework #5 Due Wednesday...

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