ECO329 Final Exam

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What does the Central Limit Theorem say?

Suppose that X1,...,Xn form a random sample from a distribution for which the mean is μ and for which the variance σ^2>0 is finite. Then when n is very large, the distribution of the sample average X1+⋯Xn/n is approximately normal with mean μ and variance σ^2/n.

What is the outcome if you type > sqrt(64)==9 | factorial(7)==5040

TRUE

Suppose that a continuous random variable 𝑋 has the following probability density function: f(x) = 3x^2 for 0≤x≤1, 0 otherwise. Calculate the expectation of X

3/4

Suppose that a student has a 40% chance of getting an A on her Psychology midterm and an 80% chance of getting an A on her Biology midterm. If those events are negatively correlated, which of the following is a mathematically possible value for the probability that the student gets an A on both midterms?

30%

Suppose that the state space of an experiment is S={1,2,3,4,5,6,7,8,9,10}. Which of the following is a partition of S?

B1={1,4,7}, B2={2,5,8}, B3={3,6,9}, B4={10}

> prior <- c(.15, .35, .4, .1)> pass <- c(1, .9, .8, .5)> z <- prior*pass> z[3] / sum(z)[1] 0.3832335

X has a finite or countable set of possible values.

Consider the variable "cty" (miles per gallon in the city) from the data set "mpg" that is included as part of the "ggplot2" package in RStudio. Suppose that we want to create a variable "ctycat" that indicates whether the value of cty is "High" (above 20), "Medium" (between 15 and 20), or "Low" (below 15). How could we do that in RStudio?

ctycat <-ifelse(mpg$cty>20,"High",ifelse(mpg$cty>15,"Medium","Low"))

Suppose that the c.d.f. F of a random variable X is given by F(x)={0for x<0x2for 0≤x≤11for x>1. What is the p.d.f. f of X?

f(x) = {2x for 0<x<1

Suppose that a random variable X has the uniform distribution on the interval [−6,4]. What is the p.d.f. f of X?

f(x)={1/10 for -6<x<4 0 otherwise}

What should you type in RStudio if you want to find out what your current working directory is?

getwd()

Suppose we want to run an experiment 3 times where the core experiment is to draw 4 numbers out of the set of integers between 1 and 10 without replacement. How could we do that in RStudio?

replicate(3,sample(1:10,4))

Suppose that you have a data frame named "example" in R. What command would you use to find out what types of elements it contains?

str(example)

Consider the data frame mpg that is included as part of the ggplot2 package in RStudio. Suppose that you want to create a subset with only the cars from 2008, and you only want to keep the variables, "model," "cyl", and "trans". What command would you use?

mpgHW5 <- subset(mpg,subset=year==2008,select=c("model","cyl", "trans"))

Suppose that a random variable X has the binomial distribution with parameters n and p. What is the expectation of X?

Suppose that the random variable X has the normal distribution with mean 1 and variance 9. What is Pr(X≤4)?

phi(1) = 0.8413

Suppose that the random variable X has the normal distribution with mean 1 and variance 9. What is Pr(−2≤X<2.5)?

phi(1/2) - phi(-1) = 0.5328

Suppose that X1,...,Xn form a random sample from the normal distribution with mean μ and variance σ2. What is the distribution of the random variable nσ′(X¯n−μ), where X¯n is the sample mean and σ′=[1n−1∑i=1n(Xi−X¯n)2]12?

t distribution with n-1 degrees of freedom.

Suppose that one card is to be drawn from a deck of 6 cards that contains 3 red cards numbered from 1 to 3 and 3 yellow cards numbered from 4 to 6. Let A be the event that the card drawn is an even number. Let B be the event that the card drawn is yellow. Let C be the event that the card drawn is a number less than 5. What is the event A∪B∪C?

{1,2,3,4,5,6}

Suppose that one card is to be drawn from a deck of 3 red cards numbered from 1 to 3, and another card is to be drawn from a deck of 3 yellow cards numbered from 4 to 6. What is the sample space S ?

{14,15,16,24,25,26,34,35,36}

Suppose that one card is to be drawn from a deck of 3 red cards numbered from 1 to 3, and another card is to be drawn from a deck of 3 yellow cards numbered from 4 to 6. Let A be the event that the red card drawn is an odd number. Which is the correct mathematical definition of A?

{14,15,16,34,35,36}

{24,25,26}

A deck contains five cards: the 3, the 7, the Jack, the Queen, and the King. You draw cards from the deck in random order without replacement until a face card (J, Q, or K) is drawn. For example, 73J is one possible sequence of draws (outcome). What is the event that the King is drawn?

{K,3K,7K,37K,73K}

Suppose that there are 12 random variables X1,...,X12, where Xi has the normal distribution with mean μi and variance σi2 (i=1,...,12). What is the expectation of the sum X1+...+X12?

Σ12, i=1 μi

In a particular pharmaceutical trial, 1/5 of the subjects are under 18 years old, 2/5 are over 65 years old, and the rest are between 18 and 65 years old. 20% of the under-18 subjects suffered a negative reaction to the drug, 10% of the 18-65 year-olds had a negative reaction, and 30% of the over-65 subjects had a negative reaction. If a given subject suffered a negative reaction, what is the probability that the subject is between 18 and 65 years old?

0.2

Suppose that a random variable X has the uniform distribution on the interval [−6,4]. What is the value of Pr(0<X<2)?

0.2

{4,5,6}

{4}

{J,Q,K,3J,3Q,3K}

0.4

A random student from ECO 329 is selected. Let A be the event that the student has seen the movie Frozen, and let B be the event that the student has seen the movie Star Wars. Suppose that Pr(A)=0.9 and that Pr(B)=0.5. The probability that the student has seen both movies is 0.495. If we know that the student has seen Frozen, then what is the probability that the student has also seen Star Wars?

0.55

A random student from ECO 329 is selected. Let A be the event that the student has seen the movie Frozen, and let B be the event that the student has seen the movie Star Wars. Suppose that Pr(A)=0.9 and that Pr(B)=0.5. The probability that the student has seen both movies is 0.495. If we know that the student has seen at least one of the two movies, then what is the probability (to 3 decimal places) that the student has seen Star Wars?

0.552

A random student from ECO 329 is selected. Let A be the event that the student has seen the movie Frozen, and let B be the event that the student has seen the movie Star Wars. Suppose that Pr(A)=0.9 and that Pr(B)=0.5. If the probability that the student has seen both movies is 0.48, then what is the probability that the student has seen at least one of the movies?

0.92

Suppose that the random variable X has the t distribution with 20 degrees of freedom. Use RStudio to calculate the probability that X≤1.724718. What is that probability?

0.950

A random student from ECO 329 is selected. Let A be the event that the student has seen the movie Frozen, and let B be the event that the student has seen the movie Star Wars. Suppose that Pr(A)=0.9 and that Pr(B)=0.5. The probability that the student has seen both movies is 0.495. If we know that the student has seen Star Wars, then what is the probability that the student has also seen Frozen?

0.99

Suppose that a continuous random variable X has the following probability density function: f(x)={4x3for 0≤x≤10otherwise. What is the 0.75 quantile of the distribution of X?

(3/4)^1/4

Suppose that a random variable X has the uniform distribution on the interval [−6,4]. What is the mean of X?

-1

Suppose that the random variable X1 has the normal distribution with mean 7 and variance 16, and that the random variable X2 has the normal distribution with mean −1 and variance 4. The covariance of X1 and X2 is −4. What is the correlation ρ(X1,X2) of X1 and X2?

-1/2

Suppose that a random variable X has the uniform distribution on the interval [−6,4]. What is the value of Pr(X=1)?

A random student from ECO 329 is selected. Let A be the event that the student has seen the movie Frozen, and let B be the event that the student has seen the movie Star Wars. Suppose that Pr(A)=0.9 and that Pr(B)=0.5. If the probability that the student has seen both movies is 0.48, then what is the probability that the student has seen Star Wars but not Frozen?

0.02

Suppose that 10,000 tickets are sold in one lottery and 5000 tickets are sold in another lottery. If a person owns 100 tickets in each lottery, what is the probability that she will win at least one first prize?

0.0298

A random variable 𝑋 takes on value 0 with probability ½, value 2 with probability 1/3, and value 4 with probability 1/6. What is the variance of X?

20/9

Suppose that a random variable X has the binomial distribution with parameters n=20 and p=0.25. How would you use R to find Pr(X=8)?

> .25^8 * .75^12 * factorial(20) / (factorial(12)*factorial(8))[1] 0.06088669

Suppose that the probability that a randomly selected worker at a particular firm will suffer an on-the-job injury is 0.01. If the firm has 10 workers, and the probabilities of an injury are independent across workers, how would you use R to calculate the probability that exactly one worker will suffer an injury?

> 10*0.01*(0.99)^9 [1] 0.09135172

Suppose that a random variable X has the binomial distribution with parameters n=10 and p=0.7. How would you use R to find all the values of k such that Pr(X=k)>0.2?

> n <- c(0:10)> n[ .2 < .7^(n) * .3^(10-n) * factorial(10) / (factorial(10-n)*factorial(n))][1] 6 7 8

Suppose that a random variable X has the binomial distribution with parameters n=20 and p=0.25. How would you use R to find Pr(X<9)?

> n <- c(0:8)> pn <- .25^(n) * .75^(20-n) * factorial(20) / (factorial(20-n)*factorial(n))> sum(pn)[1] 0.9590748

Suppose that a random sample X1,...,Xn is to be taken from the uniform distribution on the interval [0,θ] and that θ is unknown. What is the cdf Fn(1)(c) of the sample maximum, Xn(1)=max{X1,...,Xn}, for c between 0 and θ?

F(1)n (c) = (c/Θ)^n

Suppose that a random variable X has the uniform distribution on the interval [−6,4]. What is the c.d.f. F of X?

F(x)={0 for x<-6 {x+6/10 for -6<x<4 {1 for x>4

What does it mean for a continuous random variable X to have a probability density function f?

For any interval of real numbers, the probability that the value of X lies in that interval is found by taking the integral of f over that interval.

What does it mean for a random variable X to have a cumulative distribution function F?

For any real number x, Pr(X≤x)=F(x).

A random sample of 75 observations is taken from the normal distribution with mean 34 and variance 19. Which of the following commands in RStudio will tell us the value of 𝑐 0 such that the probability that the sample average 𝑋തହ lies within 𝑐 of 34 is 0.98? I. qnorm(.99,mean=34,sd=sqrt(19/75)) - 34 II. qnorm(.99)*sqrt(19/75) III. qnorm(.99,mean=34,sd=sqrt(19/75)) IV. qnorm(.99,sd=sqrt(19/75))

I, II, and IV only.

Which of the following commands could we use in RStudio in order to draw a sample of size 100 from the uniform distribution on [0,1] and calculate the sample variance? I. var(replicate(100,mean(runif(10)))) II. var(runif(100)) III. mean(replicate(100,var(runif(1))))

II only.

Suppose that the random variables X1,...,X10 are independent, and that each has the normal distribution with mean 80 and variance 3. What is the distribution of the sample average X1+⋯X10/10?

Normal dist. with mean 80 and variance 3/10

Suppose that the random variable X1 has the normal distribution with mean 7 and variance 4, and that the random variable X2 has the normal distribution with mean −1 and variance 2. The random variables X1 and X2 are independent. What is the distribution of X1+X2?

Normal distribution with mean 6 and variance 6.

In R, what is the main difference between a matrix and a data frame?

All the elements of a matrix must be of the same type, but the elements of a data frame may have elements of different type (e.g., some numeric and others text).

A random student from UT is selected. Let 𝐴 be the event that the student is an Economics major, and let 𝐵 be the event that the student is currently enrolled in ECO 329. How do we write the event that the student is an Economics major but is not enrolled in ECO 329?

A∩Bc

Binomial with parameters n=4 and p.

What does the Law of Large Numbers say?

Suppose that X1,...,Xn form a random sample from a distribution for which the mean is μ and for which the variance is finite. Then when n is very large, the probability that the sample average X1+⋯Xn/n is close to μ is close to 1.

What does the Central Limit Theorem say?

Suppose that X1,...,Xn form a random sample from a distribution for which the mean is μ and for which the variance σ2>0 is finite. Then when n is very large, Pr(X¯n−μσn≤c)≈Φ(c), where X¯n=X1+⋯Xnn is the sample mean and Φ denotes the c.d.f. of the standard normal distribution.

A soccer team will play 4 matches this season. Suppose that the team's probability of winning any given match is given by 𝑝∈[0,1] and that winning a given match is independent of winning other matches. Let the random variable 𝑋 equal the team's total number of victories this season. Suppose that the team's coach gets a bonus of $60 for each of the team's victories. Let the random variable 𝑌 equal the total amount of bonus payment to the coach for the season. What is the expectation of 𝑌? How would the answer to the previous question change if the outcomes of the 4 matches were NOT independent?

The answer would not change.

What is the "power function" of a test?

The power function specifies, for each possible value of the relevant parameter, the probability that the test will reject the null hypothesis.

What is the "critical region" of a hypothesis test?

The set of sample outcomes at which we will reject the null hypothesis.

Suppose that a continuous random variable X has the following probability density function: f(x)={4x3for 0≤x≤10otherwise. What is the median of the distribution of X?

(1/2)^1/4

What is the variance of the uniform distribution on [0,1]?

1/12

1-(.99)^10 [1] 0.09561792

Suppose that the random variable X has the normal distribution with mean 1 and variance 9. What is Pr(X>7)?

1-phi(2) = 0.0227

Suppose that X1 has the uniform distribution on the interval [0,1], and that X2 has the uniform distribution on the interval [100,101]. What are the variances of X1 and X2, respectively?

1/12 & 1/12

What is the variance of the sample mean of 10 draws from the uniform distribution on [0,2]?

1/30

If a die is rolled three times, what is the probability that the same number will show up twice in a row (for example, 112 or 633 or 555)?

11/36

240p

If a random sample of 17 observations is taken from the normal distribution with mean 50 and variance 68, what is the probability that the sample mean will lie in the interval [49.5, 50.5]?

2phi(0.25)-1=0.20

If a random sample of 68 observations is taken from the normal distribution with mean 50 and variance 17, what is the probability that the sample mean will lie in the interval [49.5, 50.5]?

2phi(1)-1=0.68

If a random sample of 32 observations is taken from the normal distribution with mean μ and variance 8, what is the probability that the sample mean will be within 0.75 of μ?

2phi(1.5)-1 = 0.3830

Suppose that a random sample of 16 observations is taken from the normal distribution with mean μ and variance 64. If μ=200, what is the probability that the sample mean will lie in the interval [197,203]?

2Φ(1.5) - 1 = 0.8664

In a particular class, 1/4 of the students are math majors, 1/2 of the students are physics majors, and 1/4 are art history majors. 30% of the math majors made an A on the midterm exam, 50% of the physics majors made an A, and 70% of the art history majors made an A. If a given student made an A on the midterm, what is the probability that the student is an art history major?

35%

A random variable 𝑋 takes on value 0 with probability ½, value 2 with probability 1/3, and value 4 with probability 1/6. What is the expectation of X?

4/3

Suppose that a continuous random variable X has the following probability density function: f(x)={4x3for 0≤x≤1 0 otherwise. What is the expectation of X?

4/5

Suppose that a student has a 60% chance of getting an A on her Psychology midterm and a 70% chance of getting an A on her Biology midterm. If those events are negatively correlated, which of the following is a mathematically possible value for the probability that the student gets an A on both midterms?

40%

If five dice are rolled, what is the probability that each of the five numbers that appear will be different?

5/54

Suppose that a continuous random variable X1 has expectation 7 and variance 4, and that a discrete random variable X2 has expectation −1 and variance 2. The correlation of X1 and X2 is 0.12. What is the expectation of X1+X2?

There are 10 different movies available to watch on a long airplane flight. If first you pick a movie, and then your friend picks a movie, and then your other friend picks a movie, how many possible sequences of three movies are possible without watching the same movie more than once?

720

Suppose that 15% of the students in a class studied 10 hours for a midterm exam, 35% studied 5 hours, 40% studied 3 hours, and 10% did not study at all. The probability that a student who studied 10 hours could pass the exam is 100%; the probability for a student who studied 5 hours is 90%; the probability for a student who studied 3 hours is 80%; and the probability for a student who did not study at all is 50%. A student is randomly selected from the class. How would you use R to answer the following question: if she passes the exam, what is the (posterior) probability that she studied 3 hours?

> prior <- c(.15, .35, .4, .1)> pass <- c(1, .9, .8, .5)> z <- prior*pass> z[3] / sum(z)[1] 0.3832335

Suppose that the weights in kilograms of the 4 fish that you caught today are 4.28, 7.14, 4.18, and 3.78. How would you use R to calculate the mean of the weights of the fish that are under 5 kg?

> weight.fish <- c(4.28,7.14,4.18,3.78) > mean(weight.fish[weight.fish<5]) [1] 4.08

Suppose that the weights in kilograms of the 7 fish that you caught today are 2.48, 1.74, 8.14, 4.29, 2.37, 3.78, and 4.41. How would you use R to calculate the mean and variance of those weights?

> weight.fish <-c(2.48, 1.74, 8.14, 4.29, 2.37, 3.78, 4.41) > mean(weight.fish) [1] 3.887143 > var(weight.fish) [1] 4.57099

Suppose that the weights in kilograms of the 7 fish that you caught today are 2.48, 1.74, 8.14, 4.29, 2.37, 3.78, and 4.41. How would you use R to calculate the mean of the weights of the fish that are over 3 kg?

> weight.fish <-c(2.48, 1.74, 8.14, 4.29, 2.37, 3.78, 4.41) >mean(weight.fish[weight.fish>3]) [1] 5.155

What is a type II error?

A decision not to reject a false null hypothesis.

What is a type I error?

A decision to reject a true null hypothesis.

What is a statistic?

A statistic is a random variable, representing a summary of the observable random variables of interest.

Suppose that a random sample of 25 observations is taken from the normal distribution with mean μ and variance 225. Which of the following commands in RStudio will tell us the value of c>0 such that the probability that the sample average X¯25 lies within c of μ is 0.99? I. qnorm(.995,sd=sqrt(225/25)) II. qnorm(.995)*sqrt(225/25) III. qnorm(.99),sd=sqrt(225/25))

I and II only.

Again suppose that we want to use RStudio to graphically represent the Central Limit Theorem. In particular, the CLT predicts that for large values of n, the binomial distribution with parameters n and p looks approximately like the normal distribution with mean np and variance n⋅p⋅(1−p). Which of the following commands in RStudio will plot the probabilities of the binomial distribution with parameters n=1000 and p=0.4, alongside the p.d.f. of the N(400,240) distribution? I. > curve(dbinom(x,1000,0.4),xlim=c(300,500))> curve(dnorm(x,mean=400,sd=sqrt(240)),xlim=c(300,500),add=T)II. > curve(dbinom(x,1000,0.4),xlim=c(300,500))> curve(400+dnorm(x,sd=sqrt(240)),xlim=c(300,500),add=T)III. > curve(dbinom(x,1000,0.4),xlim=c(300,500))> curve(dnorm(x-400,sd=sqrt(240)),xlim=c(300,500),add=T)

I and III only.

Suppose that a random sample of 16 observations is taken from the normal distribution with mean μ and variance 64. If μ=200, which of the following commands in RStudio will tell us the probability that the sample mean will lie in the interval [197,203]? I. pnorm(203,mean=200,sd=sqrt(64/16)) - pnorm(197,mean=200,sd=sqrt(64/16)) II. pnorm((203-200)/sqrt(64/16)) - pnorm((197-200)/sqrt(64/16)) III. pnorm((203-200),sd=sqrt(64/16)) - pnorm((197-200),sd=sqrt(64/16))

I, II, and III

Suppose that in RStudio we want to graph the probability density function of the normal distribution with mean 7 and variance 4. Which of the following commands will do that?

I, II, and III

A random sample of 34 observations is taken from the normal distribution with mean 19 and variance 75. Which of the following commands in RStudio will tell us the value of 𝑐>0 such that the probability that the sample average Xbar34 lies within 𝑐 of 19 is 0.98? I. qnorm(.99,sd=sqrt(75/34)) II. qnorm(.99,mean=19,sd=sqrt(75/34)) - 19 III. qnorm(.99)*sqrt(75/34) IV. qnorm(.99,mean=19,sd=sqrt(75/34))

I, II, and III only.

Suppose that we want to use RStudio to graphically represent the Central Limit Theorem. In particular, the CLT predicts that the sample mean of 100 independent draws of a Bernoulli(0.5) random variable should have a distribution that is approximately normal with mean 0.5 and variance 0.25/100. Which of the following commands in RStudio will plot the outcomes from 1000 such sample means, alongside the p.d.f. of the N(0.5,.0025) distribution? I. > hist(replicate(1000,mean(rbinom(100,1,0.5))),breaks=20,xlim=c(0,1),probability=T)> curve(pnorm(x,mean=0.5,sd=.05),xlim=c(0,1),add=T) II. > hist(replicate(100,mean(rbinom(1000,1,0.5))),breaks=20,xlim=c(0,1),probability=T)> curve(dnorm(x,mean=0.5,sd=.05),xlim=c(0,1),add=T) III. > hist(replicate(1000,mean(rbinom(100,1,0.5))),breaks=20,xlim=c(0,1),probability=T)> curve(dnorm(x,mean=0.5,sd=.05),xlim=c(0,1),add=T)

III only.

Which of the following commands could we use in RStudio in order to 1) create a sample of 100 observations of the following: the sample mean of 10 draws from the uniform distribution on [0,2]; and 2) calculate the sample variance of those 100 observations? I. replicate(var(100,mean(runif(10,max=2)))) II. var(replicate(100,mean(runif(10)))) III. var(replicate(100,mean(runif(10,max=2))))

III only.

A random sample X1,...,Xm is drawn from a distribution F that has mean mew and variance sigma squared. What do we know about the distribution of the sample mean Xbarn = 1/nΣXi?

It has mean mew and variance sigma squared over n but we do not know its exact distribution.

Suppose that X1,...,Xn form a random sample from the normal distribution with mean μ and variance σ2. If a random interval (A,B) is a 95% confidence interval for μ, what does that mean?

Pr(A<m<B)>0.95

A random variable 𝑋 has mean 𝜇. If a random interval ሺ𝐴,𝐵ሻ is a 95% confidence interval for 𝜇, what does that mean?

Pr(A<𝝁<B)≥0.95

Suppose that the random variable X has the normal distribution with mean 103 and variance 4. What is the smallest value of c such that Pr(103−c≤X<103+c)≥0.9?

c = 3.29

Suppose that the random variable X has the normal distribution with mean 3 and variance 4. What is the smallest value of c such that Pr(3−c≤X<3+c)≥0.9?

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