Practice Problems for Midterm 1 PDF

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Practice Problems for Midterm 1 Multiple Choice Questions Chapter 2 1)

The probability of an outcome a. b. c. d.

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

Answer: c 2)

The probability of an event A or B (Pr(A or B)) to occur equals a. Pr(A)  Pr(B). b. Pr(A) + Pr(B) if A and B are mutually exclusive. Pr( A) c. . Pr( B) d. Pr(A) + Pr(B) even if A and B are not mutually exclusive. Answer: b

3)

The cumulative probability distribution shows the probability e. f. g. h.

4)

that a random variable is less than or equal to a particular value. of two or more events occurring at once. of all possible events occurring. that a random variable takes on a particular value given that another event has happened.

Answer: a The expected value of a discrete random variable a. b. c. d.

is the outcome that is most likely to occur. can be found by determining the 50% value in the c.d.f. equals the population median. is computed as a weighted average of the possible outcome of that random variable, where the weights are the probabilities of that outcome. 1

Answer: d 9)

Let Y be a random variable. Then var(Y) equals E [(Y  Y ) 2 ] . b. E [| (Y  Y ) |] . c. E [(Y  Y ) 2 ] . d. E [(Y  Y )] .

a.

Answer: c The conditional distribution of Y given X = x, Pr(Y  y | X  x ) , is

10)

a.

Pr(Y  y) . Pr( X  x ) l

b.

 Pr( X  x , Y  y) . i

i 1

c. d.

Pr( X  x ,Y  y ) . Pr(Y  y) Pr( X  x ,Y  y ) . Pr( X  x)

Answer: d The conditional expectation of Y given X, E (Y | X  x ) , is calculated as follows:

11)

k

a.

 y Pr( X  x

i

|Y  y) .

i

| X  x) .

i

i 1

b. E[ E (Y | X )]. k

c.

 y Pr(Y  y i

i 1 l

d.

 E (Y | X x ) Pr(X x ) . i

i

i 1

Answer: c 9)

Two random variables X and Y are independently distributed if all of the following conditions hold, with the exception of

2

a. b. c. d.

Pr(Y  y | X  x) Pr(Y  y ) . knowing the value of one of the variables provides no information about the other. if the conditional distribution of Y given X equals the marginal distribution of Y. E (Y ) E[ E(Y | X )] .

Answer: d 9)

The correlation between X and Y a. cannot be negative since variances are always positive. b. is the covariance squared. c. can be calculated by dividing the covariance between X and Y by the product of the two standard deviations. cov( X ,Y ) . d. is given by corr( X , Y )  var( X ) var(Y ) Answer: c

10)

Two variables are uncorrelated in all of the cases below, with the exception of a. being independent. b. having a zero covariance. c. |  XY |  X2  Y2 . d. E (Y | X ) 0 . Answer: c

11)

var( aX  bY ) 

a. b. c. d.

a 2X2  b 2Y2 . a 2X2 2 ab XY  b 2Y2 .  XY   X  Y . a 2X  b Y2 .

Answer: b 12)

To standardize a variable you a. b. c. d.

subtract its mean and divide by its standard deviation. integrate the area below two points under the normal distribution. add and subtract 1.96 times the standard deviation to the variable. divide it by its standard deviation, as long as its mean is 1.

3

Answer: a 13)

Assume that Y is normally distributed N ( , 2 ) . To find Pr( c1 Y c 2) , where c1  c2 c  , you need to calculate Pr( d1 Z d2 )  and d i  i  a. b. c. d.

 ( d 2)   ( d 1) .  (1.96)   (  1.96) .  ( d 2)  (1   ( d 1)) . 1  ( ( d 2)   ( d 1)) .

Answer: a The Student t distribution is 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. Answer: d 17)

When there are  degrees of freedom, the t distribution a. b. c. d.

can no longer be calculated. equals the standard normal distribution. has a bell shape similar to that of the normal distribution, but with “fatter” tails. equals the  2 distribution.

Answer: b 18)

The sample average is a random variable and a. b. c. d.

is a single number and as a result cannot have a distribution. has a probability distribution called its sampling distribution. has a probability distribution called the standard normal distribution. has a probability distribution that is the same as for the Y 1 ,...,Y n i.i.d. variables.

Answer: b 4

19)

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 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. Answer: d

20)

The variance of Y ,  Y2 , is given by the following formula: a.  Y2 .

Y . n  2Y c. . n Y2 . d. n b.

Answer: c 21)

The mean of the sample average Y , E (Y ) , is 1  . n Y b. Y . Y c. . n Y for n > 30. d. Y

a.

Answer: b

5

22)

In econometrics, we typically do not rely on exact or finite sample distributions because a. b. c. d.

we have approximately an infinite number of observations (think of re-sampling). variables typically are normally distributed. the covariances of Yi , Yj are typically not zero. asymptotic distributions can be counted on to provide good approximations to the exact sampling distribution.

Answer: d 23) The central limit theorem states that a. the distribution for

Y  Y becomes arbitrarily well approximated by the standard Y

normal distribution. p

b. Y  Y . c. the probability that Y is in the range Y  c becomes arbitrarily close to one as n increases for any constant c  0 . d. the t distribution converges to the F distribution for approximately n > 30. Answer: a 24)

The covariance inequality states that a. 0  X2 Y 1. b.  X2 Y  X2  Y2 . c.  X2 Y   X2  Y2 .

 2X 2 d.  X Y  2 . Y Answer: b Chapter 3 1)

An estimator is a. b. c. d.

an estimate. a formula that gives an efficient guess of the true population value. a random variable. a nonrandom number. 6

Answer: c 2)

An estimate is a. b. c. d.

3)

efficient if it has the smallest variance possible. a nonrandom number. unbiased if its expected value equals the population value. another word for estimator.

Answer: b An estimator ˆ Y of the population value Y is consistent if p

a. ˆY  Y . b. its mean square error is the smallest possible. c. Y is normally distributed. d.

p

Y  0.

Answer: a

4)

An estimator ˆ Y of the population value Y is more efficient when compared to another estimator Y , if a. E( ˆ Y ) > E( Y ). b. it has a smaller variance. c. its c.d.f. is flatter than that of the other estimator. d. both estimators are unbiased, and var( ˆ Y ) < var( Y ).

Answer: d 5) The standard error of Y , SE (Y ) ˆ Y is given by the following formula: 1 n (Yi  Y )2 . i.  n i1 sY2 . n k. s Y . sY l. . n j.

Answer: d 7

7)

When you are testing a hypothesis against a two-sided alternative, then the alternative is written as a. E (Y )   Y ,0 . b. E (Y )  Y ,0 . c. Y  Y ,0 . d. E (Y )  Y ,0 . Answer: d

8)

A scatterplot a. shows how Y and X are related when their relationship is scattered all over the place. b. relates the covariance of X and Y to the correlation coefficient. c. is a plot of n observations on Xi and Yi , where each observation is represented by the point ( X i , Yi ). d. shows n observations of Y over time. Answer: c

9) The following types of statistical inference are used throughout econometrics, with the exception of a. b. c. d.

confidence intervals. hypothesis testing. calibration. estimation.

Answer: c 10)

Among all unbiased estimators that are weighted averages of Y1 ,..., Yn , Y is a. b. c. d.

the only consistent estimator of Y . the most efficient estimator of Y . a number which, by definition, cannot have a variance. the most unbiased estimator of Y .

Answer: b

8

11)

To derive the least squares estimator Y , you find the estimator m which minimizes n

e.

 (Y  m )

2

i

.

i 1 n

f. |  (Yi  m ) | . i 1 n

g.

 mY

i

2

.

1 i n

h.

 (Y  m ) i

.

1 i

Answer: a 12)

If the null hypothesis states H 0 : E (Y ) Y ,0 , then a two-sided alternative hypothesis is e. H1 : E( Y )  Y ,0 . f. H1 : E( Y )  Y ,0 . g. H1 : Y  Y ,0 . h. H1 : E( Y )  Y ,0 . Answer: a

14)

A large p-value implies e. f. g. h.

rejection of the null hypothesis. a large t-statistic. a large Y act . that the observed value Y act is consistent with the null hypothesis.

Answer: d 15)

The formula for the sample variance is 1 n  (Yi  Y ) . n  1 i1 1 n 2 s  (Yi  Y )2 . b. Y   n 1 i1 1 n (Yi   Y )2 . c. sY2    n 1 i1

a. sY2 

9

2 d. sY 

1 n 1  (Y  Y )2 . n  1 i 1 i

Answer: b 16)

Degrees of freedom a. in the context of the sample variance formula means that estimating the mean uses up some of the information in the data. b. is something that certain undergraduate majors at your university/college other than economics seem to have an  amount of. c. are (n-2) when replacing the population mean by the sample mean. d. ensure that s 2Y  2Y . Answer: a

17)

The t-statistic is defined as follows:

a.

e. f. g.

Y  Y,0 t Y2 . n Y  Y,0 t . SE(Y ) (Y  Y ,0 ) 2 t . SE (Y ) 1.96.

Answer: b 18)

The power of the test e. is the probability that the test actually incorrectly rejects the null hypothesis when the null is true. f. depends on whether you use Y or Y 2 for the t-statistic. g. is one minus the size of the test. h. is the probability that the test correctly rejects the null when the alternative is true. Answer: d

19)

The sample covariance can be calculated in any of the following ways, with the exception of:

10

1 n  (X  X )(Yi  Y ) . n  1 i 1 i n 1 n X iYi  XY . b.  n  1 i 1 n 1 1 n ( X i   X )(Yi  Y ) . c. n i1 d. r XY s Y s Y , where r XY is the correlation coefficient. a.

Answer: c

20)

When the sample size n is large, the 90% confidence interval for Y is a. Y 1.96 SE(Y ) . b. Y 1.64 SE(Y ) . c. Y 1.64 Y . d. Y 1.96 . Answer: b

21)

The standard error for the difference in means if two random variables M and W , when the two population variances are different, is a. b.

sM2  sW2 . nM  nW sM sW  . nM nW

c.

s2 1 sM2  W). ( 2 nM nW

d.

s M2 sW2  . nM nW

Answer: d 22) The following statement about the sample correlation coefficient is true. 11

a. –1 rXY 1. p

2 b. rXY  corr( X i , Yi ) . c. | rXY | 1.

d. rXY 

s2XY . s2X sY2

Answer: a

23)

The correlation coefficient a. b. c. d.

lies between zero and one. is a measure of linear association. is close to one if X causes Y. takes on a high value if you have a strong nonlinear relationship.

Answer: b

Chapter 4

1)

When the estimated slope coefficient in the simple regression model, ˆ1 , is zero, then a. b. c. d.

R2 = Y . 0 < R2 < 1. R2 = 0. R2 > (SSR/TSS).

Answer: c 2)

Heteroskedasticity means that a. b. c. d.

homogeneity cannot be assumed automatically for the model. the variance of the error term is not constant. the observed units have different preferences. agents are not all rational.

Answer: b 3)

With heteroskedastic errors, the weighted least squares estimator is BLUE. You should 12

use OLS with heteroskedasticity-robust standard errors because a. b. c. e.

this method is simpler. the exact form of the conditional variance is rarely known. the Gauss-Markov theorem holds. your spreadsheet program does not have a command for weighted least squares.

Answer: b 4)

Which of the following statements is correct? a. b. c. d.

TSS = ESS + SSR ESS = SSR + TSS ESS > TSS R2 = 1 – (ESS/TSS)

Answer: a 5)

Binary variables a. b. c. d.

are generally used to control for outliers in your sample. can take on more than two values. exclude certain individuals from your sample. can take on only two values.

Answer: d 6)

When estimating a demand function for a good where quantity demanded is a linear function of the price, you should a. b. c. d.

not include an intercept because the price of the good is never zero. use a one-sided alternative hypothesis to check the influence of price on quantity. use a two-sided alternative hypothesis to check the influence of price on quantity. reject the idea that price determines demand unless the coefficient is at least 1.96.

Answer: b 7)

The reason why estimators have a sampling distribution is that a. b. c. d.

economics is not a precise science. individuals respond differently to incentives. in real life you typically get to sample many times. the values of the explanatory variable and the error term differ across samples.

Answer: d 13

8)

The OLS estimator is derived by a. connecting the Yi corresponding to the lowest Xi observation with the Yi corresponding to the highest Xi observation. b. making sure that the standard error of the regression equals the standard error of the slope estimator. c. minimizing the sum of absolute residuals. d. minimizing the sum of squared residuals. Answer: d

9)

Interpreting the intercept in a sample regression function is a. not reasonable because you never observe values of the explanatory variables around the origin. b. reasonable because under certain conditions the estimator is BLUE. c. reasonable if your sample contains values of Xi around the origin. d. not reasonable because economists are interested in the effect of a change in X on the change in Y. Answer: c

10)

The sample average of the OLS residuals is a. b. c. d.

some positive number since OLS uses squares. zero. unobservable since the population regression function is unknown. dependent on whether the explanatory variable is mostly positive or negative.

Answer: b

11)

The t-statistic is calculated by dividing a. b. c. d.

the OLS estimator by its standard error. the slope by the standard deviation of the explanatory variable. the estimator minus its hypothesized value by the standard error of the estimator. the slope by 1.96.

Answer: c 12)

The slope estimator, 1, has a smaller standard error, other things equal, if 14

a. b. c. d.

there is more variation in the explanatory variable, X. there is a large variance of the error term, u. the sample size is smaller. the intercept, 0, is small.

Answer: a 13)

The regression R2 is a measure of a. b. c. d.

whether or not X causes Y. the goodness of fit of your regression line. whether or not ESS > TSS. the square of the determinant of R.

Answer: b 14)

(Requires Appendix) The sample regression line estimated by OLS a. b. c. d.

will always have a slope smaller than the intercept. is exactly the same as the population regression line. cannot have a slope of zero. will always run through the point ( X , Y ).

Answer: d

15)

The confidence interval for the sample regression function slope a. can be used to conduct a test about a hypothesized population regression function slope. b. can be used to compare the value of the slope relative to that of the intercept. c. adds and subtracts 1.96 from the slope. d. allows you to make statements about the economic importance of your estimate. Answer: a

16)

If the absolute value of your calculated t-statistic exceeds the critical value from the standard normal distribution, you can a. b. c. d.

reject the null hypothesis. safely assume that your regression results are significant. reject the assumption that the error terms are homoskedastic. conclude that most of the actual values are very close to the regression line. 15

Answer: a

17)

Under the least squares assumptions (zero conditional mean for the error term, Xi and Yi being i.i.d., and Xi and ui having finite fourth moments), the OLS estimator for the slope and intercept a. b. c. d.

has an exact normal distribution for n > 15. is BLUE. has a normal distribution even in small samples. is unbiased.

Answer: d 18)

To obtain the slope estimator using the least squares principle, you divide the a. b. c. d.

sample variance of X by the sample variance of Y. sample covariance of X and Y by the sample variance of Y. sample covariance of X and Y by the sample variance of X. sample variance of X by the sample covariance of X and Y.

Answer: c 19)

To decide whether or not the slope coefficient is large or small, a. b. c. d.

you should analyze the economic importance of a given increase in X. the slope coefficient must be larger than one. the slope coefficient must be statistically significant. you should change the scale of the X variable if the coefficient appears to be too small.

Answer: a 20)

E(ui | Xi) = 0 says that a. b. c. d.

dividing the error by the explanatory variable results in a zero (on average). the sample regression function residuals are unrelated to the explanatory variable. the sample mean of the Xs is much larger than the sample mean of the errors. the conditional distribution of the error given the explanatory variable has a zero mean.
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