Question 1

A type II error

Accepted Answer

A) is typically smaller than the type I error. 
B) is the error you make when choosing type II or type I. 
C) is the error you make when not rejecting the null hypothesis when it is false. 
D) cannot be calculated when the alternative hypothesis contains an "=". 
A) is typically smaller than the type I error. 
B) is the error you make when choosing type II or type I. 
C) is the error you make when not rejecting the null hypothesis when it is false. 
D) cannot be calculated when the alternative hypothesis contains an "=".

Question 2

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

Accepted Answer

A)  $\sqrt { \frac { S _ { M} ^ { 2 } + S _ { W } ^ { 2 } } { n _ {M} + n _ { W } } }$ 
B)  $\frac { S _ { M} } { n _ {W } } + \frac { S _ { M } } { n _ { W } }$ 
C)  $\sqrt { \sqrt { \frac { 1 } { 2 } \left( \frac { S _ { M } ^ { 2 } + S _ { W} ^ { 2 } } { n _ { M } + n _ {W} } \right) } }$ 
D)  $\sqrt { \frac { S _ {M} ^ { 2 } + S _ { W} ^ { 2 } } { n _ { M } + n _ { W} } }$ 
A)  $\sqrt { \frac { S _ { M} ^ { 2 } + S _ { W } ^ { 2 } } { n _ {M} + n _ { W } } }$ 
B)  $\frac { S _ { M} } { n _ {W } } + \frac { S _ { M } } { n _ { W } }$ 
C)  $\sqrt { \sqrt { \frac { 1 } { 2 } \left( \frac { S _ { M } ^ { 2 } + S _ { W} ^ { 2 } } { n _ { M } + n _ {W} } \right) } }$ 
D)  $\sqrt { \frac { S _ {M} ^ { 2 } + S _ { W} ^ { 2 } } { n _ { M } + n _ { W} } }$

Question 3

The development office and the registrar have provided you with anonymous matches of starting salaries and GPAs for 108 graduating economics majors. Your sample contains a variety of jobs, from church pastor to stockbroker.
(a)The average starting salary for the 108 students was $38,644.86 with a standard deviation of $7,541.40. Construct a 95% confidence interval for the starting salary of all economics majors at your university/college.
(b)A similar sample for psychology majors indicates a significantly lower starting salary. Given that these students had the same number of years of education, does this indicate discrimination in the job market against psychology majors?
(c)You wonder if it pays (no pun intended)to get good grades by calculating the average salary for economics majors who graduated with a cumulative GPA of B+ or better, and those who had a B or worse. The data is as shown in the accompanying table. $\begin{array} { | c | c | c | c | } 
\hline \text { Cumulative GPA } & \begin{array} { c } 
\text { Average Earnings } \
\bar { Y }
\end{array} & \begin{array} { c } 
\text { Standard deviation } \
S _ { Y }
\end{array} & n \
\hline \text { B+ or better } & \$ 39,915.25 & \$ 8,330.21 & 59 \
\hline \text { B or worse } & \$ 37,083.33 & \$ 6,174.86 & 49 \
\hline
\end{array}$Conduct a t-test for the hypothesis that the two starting salaries are the same in the population. Given that this data was collected in 1999, do you think that your results will hold for other years, such as 2002?

Accepted Answer

(a)38,644.86 ± 1.96 × @#LAT-DLM& = 38,644.86 ± 1,4

Question 4

IQs of individuals are normally distributed with a mean of 100 and a standard deviation of 16. If you sampled students at your college and assumed, as the null hypothesis, that they had the same IQ as the population, then in a random sample of size
(a)n = 25, find Pr( $\bar { Y }$  97).
(c)n = 144, find Pr(101 < $\bar { Y }$ < 103).

Accepted Answer

(a)0.94
(b

Question 5

Some policy advisors have argued that education should be subsidized in developing countries to reduce fertility rates. To investigate whether or not education and fertility are correlated, you collect data on population growth rates (Y)and education (X)for 86 countries. Given the sums below, compute the sample correlation: $\sum _ { i = 1 } ^ { n } Y _ { i }$ = 1.594; $\sum _ { i = 1 } ^ { n } X _ { i }$ = 449.6; $\sum _ { i = 1 } ^ { n } Y _ { i } X _ { i }$ = 6.4697; $\sum _ { i = 1 } ^ { n } Y _ { i } ^ { 2 }$ = 0.03982; $\sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 }$ = 3,022.76

Accepted Answer

The answer of Some policy advisors have argued that education...

Question 6

Let p be the success probability of a Bernoulli random variable Y, i.e., p = Pr(Y = 1). It can be shown that $\hat { p }$ , the fraction of successes in a sample, is asymptotically distributed N(p, $\frac { p ( 1 - p ) } { n }$ Using the estimator of the variance of $\hat { p }$ , $\frac { \hat { p } ( 1 - \hat { p } ) } { n }$ , construct a 95% confidence interval for p. Show that the margin for sampling error simplifies to 1/ $\sqrt { n }$ if you used 2 instead of 1.96 assuming, conservatively, that the standard error is at its maximum. Construct a table indicating the sample size needed to generate a margin of sampling error of 1%, 2%, 5% and 10%. What do you notice about the increase in sample size needed to halve the margin of error? (The margin of sampling error is 1.96×SE( $\hat { p }$ ))

Accepted Answer

The 95% confidence interval for p is @#IMG-DLM&_TB

Question 7

When you perform hypothesis tests, you are faced with four possible outcomes described in the accompanying table.   "☺" indicates a correct decision, and I and II indicate that an error has been made. In probability terms, state the mistakes that have been made in situation I and II, and relate these to the Size of the test and the Power of the test (or transformations of these).

Accepted Answer

I: Pr(reject H₀ @#IMG-DLM&_TB5979_11_TB5

Question 8

IQ scores are normally distributed with an average of 100 and a standard deviation of 16. Some research suggests that left-handed individuals have a higher IQ score than right-handed individuals. To test this hypothesis, a researcher randomly selects 132 individuals and finds that their average IQ is 103.2 with a sample standard deviation of 14.6. Using the results from the sample, can you reject the null hypothesis that left-handed people have an IQ of 100 vs. the alternative that they have a higher IQ? What critical value should you choose if the size of the test is 5%?

Accepted Answer

The hypothesis is @#IMG-DLM&_TB5979_11_TB5979_11_T

Question 9

Assume that under the null hypothesis, $\bar { Y }$ has an expected value of 500 and a standard deviation of 20. Under the alternative hypothesis, the expected value is 550. Sketch the probability density function for the null and the alternative hypothesis in the same figure. Pick a critical value such that the p-value is approximately 5%. Mark the areas, which show the size and the power of the test. What happens to the power of the test if the alternative hypothesis moves closer to the null hypothesis, i.e.,, ${ } ^ { \mu } \gamma$ = 540, 530, 520, etc.?

Accepted Answer

For a given size of the test,

Question 10

Think of at least nine examples, three of each, that display a positive, negative, or no correlation between two economic variables. In each of the positive and negative examples, indicate whether or not you expect the correlation to be strong or weak.

Accepted Answer

Answers will vary by student. Students f

Question 11

The following types of statistical inference are used throughout econometrics, with the exception of

Accepted Answer

A) confidence intervals. 
B) hypothesis testing. 
C) calibration. 
D) estimation. 
A) confidence intervals. 
B) hypothesis testing. 
C) calibration. 
D) estimation.

Question 12

The standard error of $\bar { Y } , S E ( \bar { Y } ) = \hat { \sigma } \bar { Y }$ is given by the following formula:

Accepted Answer

A)  $\frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \bar { Y } \right) ^ { 2 }$ 
B)  $\frac { S _ { y } ^ { n } } { n }$ 
C)  $S _ { y }$ 
D)  $\frac { S _ { y } } { \sqrt { n } }$ 
A)  $\frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \bar { Y } \right) ^ { 2 }$ 
B)  $\frac { S _ { y } ^ { n } } { n }$ 
C)  $S _ { y }$ 
D)  $\frac { S _ { y } } { \sqrt { n } }$

Question 13

For each of the accompanying scatterplots for several pairs of variables, indicate whether you expect a positive or negative correlation coefficient between the two variables, and the likely magnitude of it (you can use a small range).
(a)   (b)   (c)   (d)

Accepted Answer

(a)Positive correlation. The actual corr

Question 14

Math SAT scores (Y)are normally distributed with a mean of 500 and a standard deviation of 100. An evening school advertises that it can improve students' scores by roughly a third of a standard deviation, or 30 points, if they attend a course which runs over several weeks. (A similar claim is made for attending a verbal SAT course.)The statistician for a consumer protection agency suspects that the courses are not effective. She views the situation as follows: H0 : $\mu _ { Y }$ = 500 vs. H1 : $\mu _ { Y }$ = 530.
(a)Sketch the two distributions under the null hypothesis and the alternative hypothesis.
(b)The consumer protection agency wants to evaluate this claim by sending 50 students to attend classes. One of the students becomes sick during the course and drops out. What is the distribution of the average score of the remaining 49 students under the null, and under the alternative hypothesis?
(c)Assume that after graduating from the course, the 49 participants take the SAT test and score an average of 520. Is this convincing evidence that the school has fallen short of its claim? What is the p-value for such a score under the null hypothesis?
(d)What would be the critical value under the null hypothesis if the size of your test were 5%?
(e)Given this critical value, what is the power of the test? What options does the statistician have for increasing the power in this situation?

Accepted Answer

(a) @#IMG-DLM&_TB5979_00_TB5979_00_TB5979_00_TB597

Question 15

At the Stock and Watson (http://www.pearsonhighered.com/stock_watson)website go to Student Resources and select the option "Datasets for Replicating Empirical Results." Then select the chapter 8 CPS data set (ch8_cps.xls)into a spreadsheet program such as Excel. For the exercise, use the first 500 observations only. Using data for average hourly earnings only (ahe)and years of education (yrseduc), produce a scatterplot with earnings on the vertical axis and education level on the horizontal axis. What kind of relationship does the scatterplot suggest? Confirm your impression by adding a linear trendline. Find the correlation coefficient between the two and interpret it.

Accepted Answer

@#IMG-DLM&_TB5979_00_TB5979_00_TB5979_00_TB5979_00

Question 16

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

Accepted Answer

A)  $\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) \left( Y _ { i } - \bar { Y } \right)$ 
B)  $\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } - \frac { n } { n - 1 } \overline { X Y }$ 
C)  $\frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \mu _ { X } \right) \left( Y _ { i } - \mu _ { Y } \right)$ 
D) rXYSYSY, where rXY is the correlation coefficient. 
A)  $\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) \left( Y _ { i } - \bar { Y } \right)$ 
B)  $\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } - \frac { n } { n - 1 } \overline { X Y }$ 
C)  $\frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \mu _ { X } \right) \left( Y _ { i } - \mu _ { Y } \right)$ 
D) rXYSYSY, where rXY is the correlation coefficient.

Question 17

With i.i.d. sampling each of the following is true except

Accepted Answer

A) E $( \bar { Y } )$ = $\mu_y$ 
B) var $( \bar { Y } )$ = $\sigma _ { y } ^ { 2 }$ /n. 
C) E $( \bar { Y } )$ < E(Y). 
D)  $Y$ is a random variable. 
A) E $( \bar { Y } )$ = $\mu_y$ 
B) var $( \bar { Y } )$ = $\sigma _ { y } ^ { 2 }$ /n. 
C) E $( \bar { Y } )$ < E(Y). 
D)  $Y$ is a random variable.

Question 18

Your packaging company fills various types of flour into bags. Recently there have been complaints from one chain of stores: a customer returned one opened 5 pound bag which weighed significantly less than the label indicated. You view the weight of the bag as a random variable which is normally distributed with a mean of 5 pounds, and, after studying the machine specifications, a standard deviation of 0.05 pounds.
(a)You take a sample of 20 bags and weigh them. Sketch below what the average pattern of individual weights might look like. Let the horizontal axis indicate the sampled bag number (1, 2, …, 20). On the vertical axis, mark the expected value of the weight under the null hypothesis, and two (? 1.96)standard deviations above and below the expected value. Draw a line through the graph for E(Y)+ 2 ${ } ^ { \sigma } { } _ { Y }$ , E(Y), and E(Y)- 2 ${ } ^ { \sigma } Y$ How many of the bags in a sample of 20 will you expect to weigh either less than 4.9 pounds or more than 5.1 pounds?
(b)You sample 25 bags of flour and calculate the average weight. What is the distribution of the average weight of these 25 bags? Repeating the same exercise 20 times, sketch what the distribution of the average weights would look like in a graph similar to the one you drew in (b), where you have adjusted the standard error of $\bar { Y }$ accordingly.
(c)For each of the twenty observations in (c)a 95% confidence interval is constructed. Draw these confidence intervals, using the same graph as in (c). How many of these 20 confidence intervals would you expect to weigh 5 pounds under the null hypothesis?

Accepted Answer

(a)On average, there should be one bag i

Question 19

Assume that two presidential candidates, call them Bush and Gore, receive 50% of the votes in the population. You can model this situation as a Bernoulli trial, where Y is a random variable with success probability Pr(Y = 1)= p, and where Y = 1 if a person votes for Bush and Y = 0 otherwise. Furthermore, let $\hat { p }$ be the fraction of successes (1s)in a sample, which is distributed N(p, $\frac { p ( 1 - p ) } { n }$ )in reasonably large samples, say for n ? 40.
(a)Given your knowledge about the population, find the probability that in a random sample of 40, Bush would receive a share of 40% or less.
(b)How would this situation change with a random sample of 100?
(c)Given your answers in (a)and (b), would you be comfortable to predict what the voting intentions for the entire population are if you did not know p but had polled 10,000 individuals at random and calculated $\hat { p }$ ? Explain.
(d)This result seems to hold whether you poll 10,000 people at random in the Netherlands or the United States, where the former has a population of less than 20 million people, while the United States is 15 times as populous. Why does the population size not come into play?

Accepted Answer

(a)Pr( @#LAT-DLM& < 0.40)= Pr(Z < @#LAT-DLM& )= Pr(Z < -1.26

Question 20

If the null hypothesis states H0 : E(Y)= µY,0, then a two-sided alternative hypothesis is

Accepted Answer

A) H1 : E(Y)? µY,0. 
B) H1 : E(Y)? µY,0. 
C) H1 : $ { \mu }_Y$ Y,0. 
D) H1 : E(Y)> µY,0. 
A) H1 : E(Y)? µY,0. 
B) H1 : E(Y)? µY,0. 
C) H1 : $ { \mu }_Y$ Y,0. 
D) H1 : E(Y)> µY,0.

A type II error

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

Think of at least nine examples, three of each, that display a positive, negative, or no correlation between two economic variables. In each of the positive and negative examples, indicate whether or not you expect the correlation to be strong or weak.

The following types of statistical inference are used throughout econometrics, with the exception of

The standard error of $\bar { Y } , S E ( \bar { Y } ) = \hat { \sigma } \bar { Y }$ is given by the following formula:

For each of the accompanying scatterplots for several pairs of variables, indicate whether you expect a positive or negative correlation coefficient between the two variables, and the likely magnitude of it (you can use a small range). (a) (b) (c) (d)

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

With i.i.d. sampling each of the following is true except

If the null hypothesis states H₀ : E(Y)= µ_Y,0, then a two-sided alternative hypothesis is

Economic Questions and Data

Review of Probability

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 3: Review of Statistics

A type II error

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

Think of at least nine examples, three of each, that display a positive, negative, or no correlation between two economic variables. In each of the positive and negative examples, indicate whether or not you expect the correlation to be strong or weak.

The following types of statistical inference are used throughout econometrics, with the exception of

The standard error of Yˉ,SE(Yˉ)=σ^Yˉ\bar { Y } , S E ( \bar { Y } ) = \hat { \sigma } \bar { Y }Yˉ,SE(Yˉ)=σ^Yˉ is given by the following formula:

For each of the accompanying scatterplots for several pairs of variables, indicate whether you expect a positive or negative correlation coefficient between the two variables, and the likely magnitude of it (you can use a small range). (a) (b) (c) (d)

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

With i.i.d. sampling each of the following is true except

If the null hypothesis states H0 : E(Y)= µY,0, then a two-sided alternative hypothesis is

Economic Questions and Data

Review of Probability

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 standard error of $\bar { Y } , S E ( \bar { Y } ) = \hat { \sigma } \bar { Y }$ is given by the following formula:

If the null hypothesis states H₀ : E(Y)= µ_Y,0, then a two-sided alternative hypothesis is