prac test 1_quant methods

Name: Section:

Quantitative Methods for MBAsFinal Exam

DO NOT OPEN THE BOOKLET YETTAKE A MINUTE TO READ THE INSTRUCTIONS

Instructions.

You have 3 hours. The maximum score is 600.You can use: pen or pencil, non-graphic calculatorYou can use a two-page, hand-written cheat-sheet (letter size).You can use the last pages as scratch paper. Some probabilities for the normal and t distributionare at the end of the booklet. You will need those to answer some of the questions.

Please, read carefully before answering.We will not answer to any question during the test.If you are done with your test, you can leave.

You are supposed to behave according to the honor codeYou cannot communicate with other students.A violation of the honor code will be prosecuted.

Now relax, take a deep breath. Another deep breath.

1. Please use the following sample to answer this question:

3 4 6 3 2

(a) (20 points) What is the mean of the sample? Show your computations.

(b) (20 points) What is the median of the sample? Show your computations.

(c) (10 points) What is the mode of the sample? Explain.

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(d) (20 points) What is the variance of the sample? Show your computations.

(e) (20 points) Suppose you have 2 samples. The first sample has mean x = 2 and variances2x = 4, and the second sample has mean y = 10 and variance s2y = 9. Which sample hasmore variability? Explain your answer

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2. Consider the 3 independent random variables

X ∼ Bernoulli(0.2)

Y ∼ Bernoulli(0.3)

S ∼ Bernoulli(0.4)

(a) (20 points) Compute the expected value of the variable W = X + S

E(W ) =

(b) (20 points) Compute the variance of the variable H = X + 2Y − 3S

V (H) =

(c) (20 points) Are X an Y identically distributed? Explain.

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3. Consider the two independent random variables X and Y , with joint distribution given in thetable below.

X1 2 6 24 52 101 307

11 0.02 0.02 0.04 0.04 0.04 0.030 0.010Y 17 0.03 0.03 0.06 0.06 0.06 0.045 0.015

23 0.04 0.04 0.08 0.08 0.08 0.060 0.02024 0.01 0.01 0.02 0.02 0.02 0.015 0.005

(a) (20 points) What is the covariance of X and Y ?

(b) (20 points) Compute the conditional probability of X = 24, given Y = 17

P (X = 24|Y = 17) =

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4. Please use the histograms below to answer the following questions. The random variable X isdistributed according to a continuous uniform distribution X ∼ U [−100, 100].

(A) (B)

(C) (D)

(a) (20 points) Suppose you collect an i.i.d. sample of size n = 1000000 fromX ∼ U [−100, 100].Which graph above represent the histogram of your sample? Explain your answer.

(b) (20 points) Suppose you collect 1000 i.i.d. samples of size n = 1000 fromX ∼ U [−100, 100],and for each sample you compute the sample mean x. Which graph above represents (ap-proximately) the histogram of the sample means? Explain your answer.

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5. (a) (20 points) Find the maximum of the function f(x) = x2 − 4x+ 3 in the interval [1, 4].

(b) (20 points) Find the interior mininum of the function f(x) = x2−4x+3 in the interval [1, 4]

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6. (20 points) Compute the partial derivatives for the function

f(x, y) = 3x2y3

7. (10 points) Compute the first derivative of the function

f(x) = 3xe3x

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8. Suppose we have two independent random variables, X and Y such that

X ∼ N (20, 32)

Y ∼ N (20, 42)

(a) (10 points) What is the probability that X is greater than Y ?

P (X > Y ) =

(b) (10 points) We collect an i.i.d. sample of size nx = 100 from X and an i.i.d. sample ofsize ny = 100 from Y . What is the probability that the sample mean of X is greater thanthe sample mean of Y ?

P (X > Y ) =

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9. Use the Excel regression output below to answer part (a), (b) and (c). We estimated theregression

price = β0 + β1size+ ε

where price is the selling price in thousands of dollars and size is the size in thousands of squarefeet for a sample of houses.

(a) (10 points) Test the hypothesis

H0 : β1 = 0

H1 : β1 6= 0

at a significance level α = 0.01. Can you reject the null? Explain your answer, indicatingwhich part of the regression output you are using to answer.

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(b) (10 points) Test the hypothesis

H0 : β1 = 15

H1 : β1 6= 15

at a significance level α = 0.05. Can you reject the null? Explain your answer

(c) (10 points) What is a 95% confidence interval for the intercept β0? Explain your answer.

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(d) (20 points) The estimator for the intercept β0 is denoted as b0. A statistician tells youthat

b0 ∼ N(β0, σ

2

(1

n+

x2

(n− 1)s2x

))Using only this information, can you say if the estimator b0 is unbiased? Explain youranswer.

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10. A playboy goes to a party in New York, where he meets 4 fabulous ladies: Carrie, Samantha,Miranda and Charlotte. He asks each of them out, independently (i.e. making sure that theydo not know he invited the others too). From his past experience, the playboy knows that theprobability of success with one lady is p = 0.3.

(a) (20 points) What is the probability that all ladies reject the invitation? Explain youranswer

(b) (20 points) What is the probability that at least one lady accepts the invitation? Explainyour answer

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(c) (20 points) You do not believe that the playboy has a probability of success of p = 0.3.You collect an i.i.d. sample of 100 ladies that the playboy asked out, and compute a sampleproportion of successes of p = 0.15. Construct the rejection region for the test

H0 : p = 0.3

H1 : p < 0.3

at a significance level of α = 0.05.

(d) (10 points) Do you reject the null? Explain

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11. Xavier and Yves are two real estate agents. X denotes the number of houses sold by Xavierin a month and Y is the number of houses sold by Yves in a month. The joint probabilitydistribution of X and Y is given by

X0 1 2

0 0.12 0.42 0.06Y 1 0.21 0.06

2 0.07 0.02 0.01

(a) (10 points) Complete the table with the missing joint probability. Explain your answer

(b) (10 points) Compute the marginal distribution of X. Explain your answer

(c) (10 points) Compute the marginal distribution of Y . Explain your answer

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(d) (10 points) Compute the expected value of X (show your computations)

(e) (10 points) Compute the variance of X (show your computations)

(f) (10 points) The covariance of X and Y is σXY = −0.15 and the variance of X + Y isV (X+Y ) = 0.56. What is the variance V (Y ) of the number of cars sold by Yves? Explainhow you get your answer

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12. The weight of cereal boxes X is normally distributed. We are interested in estimating and makeinference about the population mean µ. We collect an i.i.d. sample of 25 observations, withsample mean x = 1.2 and sample variance s2 = 1.21.

(a) (20 points) Construct the rejection region for the hypothesis test

H0 : µ = 1

H1 : µ 6= 1

at a significance level α = 0.10 and compute the value of the test statistic.

(b) (10 points) What is your decision? Do you reject?

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(c) (20 points) Construct a 90% confidence interval for µ

(d) (10 points) Did you use the central limit theorem to solve (a), (b) or (c)? Explain

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13. The discrete random variable X has distribution

x 1 2 3p(x) .3 .6 .1

Please use the table below to answer (a), (b) and (c).

sample x p(x)

(a) (10 points) Consider drawing i.i.d. samples from the distribution ofX. List all the possiblesamples of size n = 2.

(b) (10 points) Compute the sample mean of each sample you listed in (a).

(c) (10 points) Compute the probability of each sample you listed in (a), i.e. the samplingdistribution of the sample mean

(d) (10 points) Compute the expected value and variance of the sample mean.

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SCRATCH PAPER

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Standard Normal Distrib

P (Z < 1.96) = 0.975 P (Z < 1.645) = 0.950 P (Z < 2.575) = 0.995P (Z < 1.282) = 0.900 P (Z < 0.845) = 0.800 P (Z < 2.330) = 0.990

Student-t distribution

P (t15 > 1.753) = 0.05 P (t16 > 1.746) = 0.05 P (t17 > 1.740) = 0.05P (t15 > 1.341) = 0.10 P (t16 > 1.337) = 0.10 P (t17 > 1.333) = 0.10P (t15 > 2.131) = 0.025 P (t16 > 2.120) = 0.025 P (t17 > 2.110) = 0.025

P (t24 > 1.711) = 0.05 P (t25 > 1.708) = 0.05 P (t26 > 1.706) = 0.05P (t24 > 1.318) = 0.10 P (t25 > 1.316) = 0.10 P (t26 > 1.315) = 0.10P (t24 > 2.064) = 0.025 P (t25 > 2.060) = 0.025 P (t26 > 2.056) = 0.025

SCRATCH PAPER

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SCRATCH PAPER

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