Quantitative Techniques 1 - Midterm(PGP- I/ TERM- I/ AY 2015-16/All Sections)

August 14, 2015Instructors: Bhattacharya, Mukhoti, Pareek, Ranjan

Total points - 100; Maximum score - 90; Bonus - 10.Duration: 2 hoursAllowed: Calculator + the course pack spiral binderInstruction: Attempt all questions. All answers should be properly justied. No marks could beclaimed without proper justication (including MCQ).

Full Name:

Roll No.: Section:

Multiple Choice Questions

1. If P (A) = P (B), then which of the following is true.

(a) A = B

(b) A B = (c) P (A|B) = P (A)(d) P (A) + P (B) = 1

2. Two fair dice are rolled independently. What is the probability that the sum of numbers onthe two dice is odd?

(a) 0

(b) 1/4

(c) 1/2

(d) 3/4

3. Student A tosses a fair coin two times and student B tosses only one time. What is theprobability that A gets more heads then B?

(a) 1

(b) 3/4

(c) 1/2

(d) 1/4

4. Bag I contains 3 red and 4 black balls while Bag II contains 5 red and 6 black balls. One ballis drawn at random from one of the bags (chosen randomly), and it is found to be red. Findthe probability that the ball was drawn from Bag II.

(a) 33/68

(b) 35/69

(c) 34/67

(d) 35/68

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5. If X bin(5, 0.2), what is the value of P (X = 2.5)?(a) 0.2

(b) 0

(c) 0.9

(d) 1

6. If X bin(11, 0.5), which of the following is correct?(a) P (X 6) = 0.9(b) P (X 6) > 0.9(c) P (X 3) = P (X 8)(d) None of the above

7. A Mumbai based rm has only two stores; one in Mumbai and another one in Newyork. Youhave the mean and standard deviation of demand at two stores in Mumbai and Newyork.(Hint : Assume demand follows Normal distribution)

Store Location Mean Demand Std. Dev.

Mumbai 1200 160Newyork 1650 220

What is the distribution of the total demand?

(a) N(1850, 380)

(b) N(1850/2, 3802/2)

(c) N(1850, (1602 + 2202)/2)

(d) None of the above

8. Which of the following statement is correct?

(a) Z/2 = /2

(b) Z/2 = Z/2

(c) Z1/2 = Z/2(d) Z/2 is a normal quantile

9. If Y = 2 + 0.1X, nd the value of Corr(X,Y )

(a) 1

(b) 0

(c) 0.01

(d) 0.1

10. If X and Y are independent variables with V (X) = V (Y ), then Corr(X Y,X + Y ) is(a) 1

(b) 0

(c) 0.5

(d) -1

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Assuming demand at the two stores are independent

Short answer type

1. (10 points) In a town of n + 1 inhabitants, a person tells a rumor to a second person, whoin turn repeats it to a third person, etc. At each step the recipient of the rumor is chosen atrandom from the n people. Find the probability that the rumor will be told

(a) exactly r times before returning to the originator. [5]

(b) exactly r times without being repeated to any person. [5]

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2. (10 points) A student is taking a multiple choice exam in which each question has 5 possibleanswers, exactly one of which is correct. If the student knows the answer (s)he selects thecorrect answer. Otherwise (s)he selects one answer at random from the 5 possible answers.Suppose that the student knows the answer to 70% of the questions.

(a) Let X be the total number of correct answers. Is X a binomial variable? Explain. [2]

(b) Find the probability that (s)he correctly answers a given question. [3]

(c) If the exam contains 75 questions and each question is worth 2 points, what is his/herexpected score in the exam? [5]

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3. (10 points) According to Cerulli Associates of Boston, 30% of all CPA nancial advisors havean average client size between $500,000 and $1 million. Thirty-four percent have an averageclient size between $1 million and $5 million. Suppose a complete list of all CPA nancialadvisors is available and 38 are randomly selected from that list.

(a) What is the expected number of CPA nancial advisors that have an average client sizebetween $500,000 and $1million? What is the expected number with an average clientsize between $1 million and $5 million? [1+1]

(b) What is the probability that at most two of the CPA nancial advisors have an averageclient size between $500,000 and $1 million ? What is the probability that at most twohave an average client size between $1 million and $5 million? Which probability ishigher and why? [3+3+2]

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Do not deduct points if someone hasn't done the continuity correction.

The probability that "at most two are in the $500,000 to $1 million" is higher because there is a smaller percentage in thatcategory which is closer to zero.

4. (10 points) According to the UN Environmental Program and WHO, in Bombay, India, airpollution standards for particulate matter are exceeding the critical level an average of 5.75days in every three-week period. Assume that the distribution of number of days exceedingthe critical level of pollution standards is Poisson distributed.

(a) What is the probability that the pollution standard is not exceeding the critical level onany day during a three-week period? [3]

(b) What is the probability that the UNEP & WHO did not nd critical pollution level onany of the ve random visits of one-week period each? [7]

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5. (10 points) A family of pdfs (referred to as the Pareto distribution) has typically been usedto approximate the distribution of income, city population size, and size of rms. The familyhas two parameters, k and , both > 0, and the pdf is

fX(x) =

kkxk+1

x 0 x <

Find the median of this distribution.

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6. (10 points) The average length of time between arrivals at a turnpike tollbooth is 23 seconds.Assume that the time between arrivals at the tollbooth is exponentially distributed.

(a) What is the probability that a minute or more will elapse between arrivals? [4]

(b) Suppose the tollbooth operator wants to take a break, and wants to make an argumentto his boss. He wants to compute the following probability: If no vehicle has arrived atthe tollbooth in the last 13 seconds, what is the probability that no vehicle will show upfor another 77 seconds? [6]

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7. (10 points) The joint distribution of (X) family income (in thousand rupees) and (Y ) numberof children who completed primary education in a family in the rural part of the country isgiven as follows:

Y 0 1 2 TotalX

15 1/10 1/10 2/10 2/530 ? ? 1/30 ?45 11/60 ? ? ?

Total ? 23/60 17/60 1

It is noted that the average family income in the rural part is 57/2. Find the missing valuesin the joint distribution table.

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A1. (10 points) A public-interest group was planning to make a court challenge to auto-insurancerates in one of the three cities: Atlanta, Baltimore or Cleveland. The probabilities it willchoose Atlanta was 0.4; Baltimore 0.35; and Cleveland with the remaining probability. Thegroup also knew that it had a 60% chance of a favourable ruling if it chose Baltimore, 45%chance if it chose Atlanta and 35% chance if it chose Cleveland. If the group did receive afavourable ruling which city did it most likely choose?

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A2. (10 points) According to the IRS, the refunds for income tax returns for one year averaged$1,332 with a standard deviation of $725. Assume that amounts owed or due on tax returnsare normally distributed.

(a) What proportion of tax returns show a refund greater than $2000? [5]

(b) What proportion of the tax returns show a refund between $100 and $700? [5]

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