**Unformatted text preview: **Stat 104: Quantitative Methods for Economics Homework 3: Due Friday, September 25 Melissa Kaplan 1) A department store manager has monitored the numbers of complaints received per week about poor service. The probabilities for numbers of complaints in a week, established by this review, are shown in the table. Let A be the event "There will be at least one complaint in a week," and B the event "There will be less than 10 complaints in a week." NUMBER OF COMPLAINTS 0 1-‐3 4-‐6 7-‐9 PROBABILITY .14 .39 .23 .15 a)
b)
c)
d)
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f)
g) 10-‐
12 .06 More than 12 .03 Find the probability of A. 1-‐.14=.86 Find the probability of B.1-‐.06-‐.06=.91 Find the probability of the complement of A. .14 Find the probability of A or B. 1 Find the probability of A and B. .39+.23+.15=.77 Are A and B mutually exclusive? No Are A and B collectively exhaustive? Yes 2) When 2 dice are rolled, find the probability of the following situations(the sample space for this is shown below and might help you answer the question). a)
b)
c)
d)
e) A sum of 5 or 6 9/36=1/4 A sum greater than 9 1/6 A sum less than 4 or greater than 9 1/4 A sum that is divisible by 4 1/4 A sum of 14 0 1 f) A sum less than 13 1 3) A silver dollar is flipped twice. Calculate the probability of each of the following occurring: a) a head on the first flip 1/2 b) a tail on the second flip given that the first toss was a head 1/2 c) two tails 1/4 d) a tail on the first and a head on the second 1/4 e) a tail on the first and a head on the second or a head on the first and a tail on the second 1/2 f) at least one head on the two flips 3/4 4) Consider the following enrollment figures from the University of Southern North Dakota in Hoople ( ): Fresh Soph Junior Senior Male 200 150 250 200 Female 200 125 200 175 What is the probability that a student, chosen at random, will be a)
b)
c)
d) a freshman 400/1500=4/15 male 8/15 a freshman or a junior 850/1500=.567 a male or a freshman 1-‐500/1500=.67 5) The personnel department of a company has compiled the data shown in the following table. Manager Promoted Not Promoted Male 46 184 Female 8 32 a) Find P(Promoted) 1/5 b) Find P(Promoted | Male)=P(promoted&male)/P(male)=.17/.85=1/5 c) Does there seem to be any discrimination in awarding promotions? Explain. No there does not seem to be any discrimination in awarding promotions. The probability of being promoted as a male is the same as the probability of being promoted in general so therefore these two are independent. 2 6) 7) 8) 9) At Google, 70% of employees know how to program in C++, 60% know Python, and 50% know both languages. a) If an employee knows C++, what is the probability that she also knows Python? .5/.7=5/7=.714 b) If an employee knows Python, what is the probability that she also knows C++? .5/.6=5/6=.833 c) If an employee knows at least one of these languages, what is the probability that she knows both of them? P(Both languages|knows one) P(knows one)=.7+.6-‐.5=.8 So therefore .5/.8=.625 d) Are the events ”knows C++” and ”knows Python” independent? Explain. They are dependent. P(Python|C++)=5/7 and the P(python)=.6 5/7 doesn’t equal .6 so therefore they are dependent. Consider four computer firms, A, B, C, D, bidding for a certain contract. A survey of past bidding success of these firms on similar contracts shows the following probabilities of winning: P(A) = 0.35, P(B) = 0.15, P(C) = 0.3, P(D) = 0.2. Before the decision is made to award the contract, firm B withdraws its bid. Find the new probability of A winning the bid (we are looking here for P( A | B ) ). P(A|)=(.35)(.85)/.85= .35 Read the pdf document on the website entitled Birthday Problems. Then answer the following question (question 3 on page 199 of the document): A small class contains 6 students. What is the chance that at least two have the same birthmonth? 1-‐(12*11*10*9*8*7/12^6)=.78 A market research firm is investigating the appeal of three package designs. The table below gives information obtained through a sample of 200 consumers. The three package designs are labeled A, B, and C. The consumers are classified according to age and package design preference. A B C Total Under 25 years 22 34 40 96 25 or older 54 28 22 104 Total 76 62 62 200 a) If one of these consumers is randomly selected, what is the probability that the person prefers design A? 76/200=.38 b) If one of these consumers is randomly selected, what is the probability that the person prefers design A and is under 25? 22/200=.11 3 c) If one of these consumers is randomly selected and is under 25, what is the probability that the person prefers design A? .23 d) If one of these consumers is randomly selected and prefers design B, what is the probability that the person is 25 or older? P(25 and older|B)=P(25+ &B)/ p(B)= .14/.31=.45 e) Are “B” and “25 or older” independent and why or why not? They are dependent. The conditional probability of being 25 and older given B is .45 and probability of being 25 & older is .52, since these are not equal they are dependent. 10) Even though independent gasoline stations have been having a difficult time, Susan Solomon has been thinking about starting her own independent gasoline station. Susan’s problem is to decide how large her station should be. The annual returns will depend on both the size of her station and a number of marketing factors related to the oil industry and demand for gasoline. After a careful analysis, Susan developed the following table of profits: a) What is the maximax decision ? 300,000. Very large station b) What is the maximin decision? -‐10,000 small station 11) A group of medical professionals is considering the construction of a private clinic. If the medical demand is high (i.e., there is a favorable market for the clinic), the physicians could realize a net profit of $100,000. If the market is not favorable, they could lose $40,000. Of course, they don’t have to proceed at all, in which case there is no cost. In the absence of any market data, the best the physicians can guess is that there is a 50–50 chance the clinic will be successful. Construct a decision tree to help analyze this problem. What should the medical professionals do? 4 100,000 (50%) Construct -‐40,000 (50%) Private prac+ce Don’t Construct 0 12) This was supposed to be on an earlier homework but we hadn’t fully covered correlation and covariance at that point yet. Load some old class survey data into Stata using the command use Some of you may have already done this problem already-‐great, you can do it again very easily then. 100 150 weight 200 250 a) Make a scatter plot of the variables weight and length. You can use the command, scatter weight height. Does there appear to be any association between the variables? Also compute the correlation and covariance between weight and height. 50 60 70
height 80 5 The variables seem to be positively correlated. R=.67. covariance= 87.87 b) Using the standard deviations of weight and length, show how the covariance divided by the standard deviations equals the correlation value. . summarize weight height
Variable Obs weight 126 height 126 Mean Std. Dev. Min Max 147.381 26.60552 100 235 67.78571 4.938999 50 80 . covariance/sx*sy=correlation 87.87/(26.6*4.94)=.67 13) Regression fits a line to data by minimizing an objective function; that is, it finds the line (slope and intercept) that minimizes the sum of the residuals squared. There are many other types of possible objective function [one could minimize the sum of the absolute values of the residuals for example]. Because regression is so important, and this idea of an objective function useful, it is important to see how to use a computer for function optimization. Finally, what you have to do for this problem: A farmer has 2400 feet of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? a) Let x be the top length of the fence. Write a function for the area of the field as a function of x. Y=x(2400-‐2x) b) Find the maximum value [click maximum in the Solver routine] 600 c) Write up your solution clearly and include a screen shot of your spreadsheet solution. 6 7 ...

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- Spring '15
- Michael Parzen
- Probability, Melissa, Dakota, 250