Question 1

The probability of an event $A \text { or } B ( \operatorname { Pr } ( A \text { or } B ) )$
to occur equals

Accepted Answer

A)  $\operatorname { Pr } ( A ) \times \operatorname { Pr } ( B )$ 
B)  $\operatorname { Pr } ( A ) + \operatorname { Pr } ( B ) \text { if } A \text { and } B \text { are mutually exclusive. }$ 
C)  $\frac { \operatorname { Pr } ( A ) } { \operatorname { Pr } ( B ) }$ 
D)  $\operatorname { Pr } ( A ) + \operatorname { Pr } ( B ) \text { even if } A \text { and } B \text { are not mutually exclusive. }$ 
A)  $\operatorname { Pr } ( A ) \times \operatorname { Pr } ( B )$ 
B)  $\operatorname { Pr } ( A ) + \operatorname { Pr } ( B ) \text { if } A \text { and } B \text { are mutually exclusive. }$ 
C)  $\frac { \operatorname { Pr } ( A ) } { \operatorname { Pr } ( B ) }$ 
D)  $\operatorname { Pr } ( A ) + \operatorname { Pr } ( B ) \text { even if } A \text { and } B \text { are not mutually exclusive. }$

Question 2

The sample average is a random variable and a. is a single number and as a result cannot have a distribution.
b. has a probability distribution called its sampling distribution.
c. has a probability distribution called the standard normal distribution.
d. has a probability distribution that is the same as for the $Y _ { 1 } , \ldots , Y _ { n }$ i.i.d. variables.

Accepted Answer

The answer of The sample average is a random variable...

Question 3

If variables with a multivariate normal distribution have covariances that equal zero, then a. the correlation will most often be zero, but does not have to be.
b. the variables are independent.
c. you should use the $\chi ^ { 2 }$ distribution to calculate probabilities.
d. the marginal distribution of each of the variables is no longer normal.

Accepted Answer

The answer of If variables with a multivariate normal distribution...

Question 4

The table accompanying lists the joint distribution of unemployment in the United States
in 2001 by demographic characteristics (race and gender). Joint Distribution of Unemployment by Demographic Characteristics, United States, 2001
$\begin{array}{|c|c|c|c|}
\hline & \begin{array}{c}
\text { White } \
(Y=0)
\end{array} & \begin{array}{c}
\text { Black and Other } \
(Y=1)
\end{array} & \text { Total } \
\hline \begin{array}{c}
\text { Age } 16-19 \
(X=0)
\end{array} & 0.13 & 0.05 & 0.18 \
\hline \begin{array}{c}
\text { Age 20 and above } \
(X=1)
\end{array} & 0.60 & 0.22 & 0.82 \
\hline \text { Total } & 0.73 & 0.27 & 1.00 \
\hline
\end{array}$
 (a)What is the percentage of unemployed white teenagers?

Accepted Answer

The original table showed the

Question 5

The central limit theorem a. states conditions under which a variable involving the sum of $Y _ { 1 } , \ldots , Y _ { n }$ i.i.d. variables becomes the standard normal distribution.
b. postulates that the sample mean $\bar { Y }$ is a consistent estimator of the population mean $\mu _ { Y }$.
c. only holds in the presence of the law of large numbers.
d. states conditions under which a variable involving the sum of $Y _ { 1 } , \ldots , Y _ { n }$ i.i.d. variables becomes the Student $t$ distribution.

Accepted Answer

The answer of The central limit theorem a. states conditions...

Question 6

$\operatorname { var } ( a X + b Y ) =$

Accepted Answer

A)  $a ^ { 2 } \sigma _ { X } ^ { 2 } + b ^ { 2 } \sigma _ { Y } ^ { 2 }$ 
B)  $a ^ { 2 } \sigma _ { X } ^ { 2 } + 2 a b \sigma _ { X Y } + b ^ { 2 } \sigma _ { Y } ^ { 2 }$ 
C)  $\sigma _ { X Y } + \mu _ { X } \mu _ { Y }$ 
D)  $a \sigma _ { X } ^ { 2 } + b \sigma _ { Y } ^ { 2 }$ 
A)  $a ^ { 2 } \sigma _ { X } ^ { 2 } + b ^ { 2 } \sigma _ { Y } ^ { 2 }$ 
B)  $a ^ { 2 } \sigma _ { X } ^ { 2 } + 2 a b \sigma _ { X Y } + b ^ { 2 } \sigma _ { Y } ^ { 2 }$ 
C)  $\sigma _ { X Y } + \mu _ { X } \mu _ { Y }$ 
D)  $a \sigma _ { X } ^ { 2 } + b \sigma _ { Y } ^ { 2 }$

Question 7

Find the following probabilities:
(a) $Y \text { is distributed } \chi _ { 4 } ^ { 2 } \text {. Find } \operatorname { Pr } ( Y > 9.49 ) \text {. }$

Accepted Answer

The answer of Find the following probabilities:
(a) \(Y \text {...

Question 8

Think of an example involving five possible quantitative outcomes of a discrete random
variable and attach a probability to each one of these outcomes.Display the outcomes,
probability distribution, and cumulative probability distribution in a table.Sketch both
the probability distribution and the cumulative probability distribution.

Accepted Answer

Answers will vary by student.T

Question 9

The covariance inequality states that a. $\quad 0 \leq \sigma _ { X Y } ^ { 2 } \leq 1$.
b. $\sigma _ { X Y } ^ { 2 } \leq \sigma _ { X } ^ { 2 } \sigma _ { Y } ^ { 2 }$.
c. $\sigma _ { X Y } ^ { 2 } - \sigma _ { X } ^ { 2 } \leq \sigma _ { Y } ^ { 2 }$.
d. $\quad \sigma _ { X Y } ^ { 2 } \leq \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { Y } ^ { 2 } }$.

Accepted Answer

The answer of The covariance inequality states that a. \(\quad...

Question 10

The textbook mentioned that the mean of  Y, E(Y)  is called the first moment of  Y , and that the expected value of the square of $Y , E \left( Y ^ { 2 } \right)$ is called the second moment of  Y , and so on. These are also referred to as moments about the origin. A related concept is moments about the mean, which are defined as $E \left[ \left( Y - \mu _ { Y } \right) ^ { r } \right]$
What do you call the second moment about the mean? What do you think the third moment, referred to as "skewness," measures? Do you believe that it would be positive or negative for an earnings distribution? What measure of the third moment around the mean do you get for a normal distribution?

Accepted Answer

The second moment about the me

Question 11

The skewness of the distribution of a random variable Y is defined as follows: a. $\frac { E \left[ \left( Y ^ { 3 } - \mu _ { Y } \right) \right] } { \sigma _ { Y } ^ { 2 } }$
b. $E \left[ \left( Y - \mu _ { Y } \right) ^ { 3 } \right]$
c. $\frac { E \left[ \left( Y ^ { 3 } - \mu _ { Y } ^ { 3 } \right) \right] } { \sigma _ { Y } ^ { 3 } }$
d. $\frac { E \left[ \left( Y - \mu _ { Y } \right) ^ { 3 } \right] } { \sigma _ { Y } ^ { 3 } }$

Accepted Answer

The answer of The skewness of the distribution of a...

Question 12

The systolic blood pressure of females in their 20s is normally distributed with a mean of
120 with a standard deviation of 9.What is the probability of finding a female with a
blood pressure of less than 100? More than 135? Between 105 and 123? You visit the
women's soccer team on campus, and find that the average blood pressure of the 25
members is 114.Is it likely that this group of women came from the same population?

Accepted Answer

The answer of The systolic blood pressure of females in...

Question 13

Assume that  Y  is normally distributed $N \left( \mu , \sigma ^ { 2 } \right)$   To find $\operatorname { Pr } \left( c _ { 1 } \leq Y \leq c _ { 2 } \right)$   where $c _ { 1 } < c _ { 2 }$ and $d _ { i } = \frac { c _ { i } - \mu } { \sigma }$, you need to calculate $\operatorname { Pr } \left( d _ { 1 } \leq Z \leq d _ { 2 } \right) =$

Accepted Answer

A)  $\Phi \left( d _ { 2 } \right) - \Phi \left( d _ { 1 } \right)$ 
B)  $\Phi ( 1.96 ) - \Phi ( - 1.96 )$ 
C)  $\Phi \left( d _ { 2 } \right) - \left( 1 - \Phi \left( d _ { 1 } \right) \right)$ 
D)  $1 - \left( \Phi \left( d _ { 2 } \right) - \Phi \left( d _ { 1 } \right) \right)$ 
A)  $\Phi \left( d _ { 2 } \right) - \Phi \left( d _ { 1 } \right)$ 
B)  $\Phi ( 1.96 ) - \Phi ( - 1.96 )$ 
C)  $\Phi \left( d _ { 2 } \right) - \left( 1 - \Phi \left( d _ { 1 } \right) \right)$ 
D)  $1 - \left( \Phi \left( d _ { 2 } \right) - \Phi \left( d _ { 1 } \right) \right)$

Question 14

The Student t distribution is

Accepted Answer

A) the distribution of the sum of m squared independent standard normal random variables. 
B) the distribution of a random variable with a chi-squared distribution with m degrees of freedom, divided by m. 
C) always well approximated by the standard normal distribution. 
D) the distribution of the ratio of a standard normal random variable, divided by the square root of an independently distributed chi-squared random variable with m
Degrees of freedom divided by m. 
A) the distribution of the sum of m squared independent standard normal random variables. 
B) the distribution of a random variable with a chi-squared distribution with m degrees of freedom, divided by m. 
C) always well approximated by the standard normal distribution. 
D) the distribution of the ratio of a standard normal random variable, divided by the square root of an independently distributed chi-squared random variable with m
Degrees of freedom divided by m.

Question 15

The probability of an outcome

Accepted Answer

A) is the number of times that the outcome occurs in the long run. 
B) equals M× N, where M is the number of occurrences and N is the population size. 
C) is the proportion of times that the outcome occurs in the long run. 
D) equals the sample mean divided by the sample standard deviation. 
A) is the number of times that the outcome occurs in the long run. 
B) equals M× N, where M is the number of occurrences and N is the population size. 
C) is the proportion of times that the outcome occurs in the long run. 
D) equals the sample mean divided by the sample standard deviation.

Question 16

To infer the political tendencies of the students at your college/university, you sample 150 of them.Only one of the following is a simple random sample: You

Accepted Answer

A) make sure that the proportion of minorities are the same in your sample as in the entire student body. 
B) call every fiftieth person in the student directory at 9 A.m.If the person does not answer the phone, you pick the next name listed, and so on. 
C) go to the main dining hall on campus and interview students randomly there. 
D) have your statistical package generate 150 random numbers in the range from 1 to the total number of students in your academic institution, and then choose
The corresponding names in the student telephone directory. 
A) make sure that the proportion of minorities are the same in your sample as in the entire student body. 
B) call every fiftieth person in the student directory at 9 A.m.If the person does not answer the phone, you pick the next name listed, and so on. 
C) go to the main dining hall on campus and interview students randomly there. 
D) have your statistical package generate 150 random numbers in the range from 1 to the total number of students in your academic institution, and then choose
The corresponding names in the student telephone directory.

Question 17

The following problem is frequently encountered in the case of a rare disease, say AIDS, when determining the probability of actually having the disease after testing positively for HIV. (This is often known as the accuracy of the test given that you have the disease.) Let us set up the problem as follows: $Y = 0$ if you tested negative using the ELISA test for HIV, $Y = 1$ if you tested positive; $X = 1$ if you have HIV, $X = 0$ if you do not have HIV. Assume that $0.1$ percent of the population has HIV and that the accuracy of the test is $0.95$ in both cases of (i) testing positive when you have HIV, and (ii) testing negative when you do not have HIV. (The actual ELISA test is actually $99.7$ percent accurate when you have HIV, and $98.5$ percent accurate when you do not have HIV.) (a)Assuming arbitrarily a population of 10,000,000 people, use the accompanying table to
first enter the column totals. $$\begin{array} { | r | l | l | l | } 
\hline & \text { Test Positive } ( Y = 1 ) & \text { Test Negative } ( Y = 0 ) & \text { Total } \
\hline \text { HIV } ( X = 1 ) & & & \
\hline \text { No HIV } ( X = 0 ) & & & \
\hline \text { Total } & & & 10,000,000 \
\hline
\end{array}$$

Accepted Answer

Answers will vary by student.Perhaps a n

Question 18

The mean of the sample average $\bar { Y } , E ( \bar { Y } )$ is

Accepted Answer

A)  $\frac { 1 } { n } \mu _ { Y }$ 
B)  $\mu _ { Y }$ 
C)  $\frac { \mu _ { Y } } { \sqrt { n } }$ 
D)  $\frac { \sigma _ { Y } } { \mu _ { Y } }$ for  n>30 . 
A)  $\frac { 1 } { n } \mu _ { Y }$ 
B)  $\mu _ { Y }$ 
C)  $\frac { \mu _ { Y } } { \sqrt { n } }$ 
D)  $\frac { \sigma _ { Y } } { \mu _ { Y } }$ for  n>30 .

Question 19

What is the probability of the following outcomes? (a) $\quad \operatorname { Pr } ( M = 7 )$
(b) $\operatorname { Pr } ( M = 2$ or $M = 10 )$
(c) $\operatorname { Pr } ( M = 4$ or $M \neq 4 )$
(d) $\operatorname { Pr } ( M = 6$ and $M = 9 )$
(e) $\operatorname { Pr } ( M  10 )$

Accepted Answer

The answer of What is the probability of the following...

Question 20

Explain why the two probabilities are identical for the standard normal distribution:
$\operatorname { Pr } ( - 1.96 \leq X \leq 1.96 ) \text { and } \operatorname { Pr } ( - 1.96 < X < 1.96 )$

Accepted Answer

For a continuous dis

The probability of an event $A \text { or } B ( \operatorname { Pr } ( A \text { or } B ) )$ to occur equals

$\operatorname { var } ( a X + b Y ) =$

Find the following probabilities: (a) $Y \text { is distributed } \chi _ { 4 } ^ { 2 } \text {. Find } \operatorname { Pr } ( Y > 9.49 ) \text {. }$

The Student t distribution is

The probability of an outcome

To infer the political tendencies of the students at your college/university, you sample 150 of them.Only one of the following is a simple random sample: You

The mean of the sample average $\bar { Y } , E ( \bar { Y } )$ is

Explain why the two probabilities are identical for the standard normal distribution: $\operatorname { Pr } ( - 1.96 \leq X \leq 1.96 ) \text { and } \operatorname { Pr } ( - 1.96 < X < 1.96 )$

Economic Questions and Data

Review of Statistics

Linear Regression With One Regressor

Regression With a Single Regressor: Hypothesis Tests and Confidence Intervals

Linear Regression With Multiple Regressors

Hypothesis Tests and Confidence Intervals in Multiple Regression

Nonlinear Regression Functions

Assessing Studies Based on Multiple Regression

Regression With Panel Data

Regression With a Binary Dependent Variable

Instrumental Variables Regression

Experiments and Quasi-Experiments

Introduction to Time Series Regression and Forecasting

Estimation of Dynamic Causal Effects

Additional Topics in Time Series Regression

The Theory of Linear Regression With One Regressor

The Theory of Multiple Regression

Filters

Exam 2: Review of Probability

The probability of an event A or B(Pr⁡(A or B))A \text { or } B ( \operatorname { Pr } ( A \text { or } B ) )A or B(Pr(A or B)) to occur equals

var⁡(aX+bY)=\operatorname { var } ( a X + b Y ) =var(aX+bY)=

Find the following probabilities: (a) Y is distributed χ42. Find Pr⁡(Y>9.49). Y \text { is distributed } \chi _ { 4 } ^ { 2 } \text {. Find } \operatorname { Pr } ( Y > 9.49 ) \text {. }Y is distributed χ42​. Find Pr(Y>9.49).

The Student t distribution is

The probability of an outcome

To infer the political tendencies of the students at your college/university, you sample 150 of them.Only one of the following is a simple random sample: You

The mean of the sample average Yˉ,E(Yˉ)\bar { Y } , E ( \bar { Y } )Yˉ,E(Yˉ) is

Explain why the two probabilities are identical for the standard normal distribution: Pr⁡(−1.96≤X≤1.96) and Pr⁡(−1.96<X<1.96)\operatorname { Pr } ( - 1.96 \leq X \leq 1.96 ) \text { and } \operatorname { Pr } ( - 1.96 < X < 1.96 )Pr(−1.96≤X≤1.96) and Pr(−1.96<X<1.96)

Economic Questions and Data

Review of Statistics

Linear Regression With One Regressor

Regression With a Single Regressor: Hypothesis Tests and Confidence Intervals

Linear Regression With Multiple Regressors

Hypothesis Tests and Confidence Intervals in Multiple Regression

Nonlinear Regression Functions

Assessing Studies Based on Multiple Regression

Regression With Panel Data

Regression With a Binary Dependent Variable

Instrumental Variables Regression

Experiments and Quasi-Experiments

Introduction to Time Series Regression and Forecasting

Estimation of Dynamic Causal Effects

Additional Topics in Time Series Regression

The Theory of Linear Regression With One Regressor

The Theory of Multiple Regression

Filters

The probability of an event $A \text { or } B ( \operatorname { Pr } ( A \text { or } B ) )$ to occur equals

$\operatorname { var } ( a X + b Y ) =$

Find the following probabilities: (a) $Y \text { is distributed } \chi _ { 4 } ^ { 2 } \text {. Find } \operatorname { Pr } ( Y > 9.49 ) \text {. }$

The mean of the sample average $\bar { Y } , E ( \bar { Y } )$ is

Explain why the two probabilities are identical for the standard normal distribution: $\operatorname { Pr } ( - 1.96 \leq X \leq 1.96 ) \text { and } \operatorname { Pr } ( - 1.96 < X < 1.96 )$