US Statistics Manufacturer Worksheet

Questions requiring numerical/written answers1. A manufacturer has accumulated the following information from its
two factories
Sample mean
Sample standard deviation
Sample size
Daily defects in Sydney
4000
100
100
Daily defects in Melbourne
4300
70
100
The manager of the Melbourne factory claims there is no significant difference in the
number of defects in the two factories. Test the manager’s claim at 1% level of
significance (assume that the population are normally distributed and have equal
variances).
2. Referring to the above table, what is the estimated standard error of the difference
between the two sample means? (Assume that the two population variances are equal)
3.Given the following information on two independent samples calculate a 95% confidence
interval. Assume the populations have equal variances
Introductory Econometrics
Problem Set 1
Assignment instructions
•
•
•
•
•
•
•
You must submit your work via the Turnitin link on moodle by 18:00 on Friday
March 11.
This assignment will be marked for the course assessment and will be worth 7.5% of
your final mark. You must attach your Stata do-file to your answers (by typing
Stata codes or pasting screenshots of the do-file window or the result window in
your answer) and failure to do so will result in a zero mark for the computing
questions.
Name, student number, course title, tutorial group number and tutor’s name should be
clearly included in the submission. Your answers including Stata do-file should not
exceed 6 pages. The Assignment is based on the material covered in both lectures and
tutorials up to the end of Week 3.
The assignment is INDIVIDUAL work. You may discuss the assignment with your
peers, but you must submit YOUR OWN answers.
If the answer requires some mathematical calculation, show the steps, don’t just report
the final results.
This assignment has a total of 100 points awarded.
All submissions may be checked for plagiarism. The University regards plagiarism as
a form of academic misconduct and has very strict rules regarding plagiarism. For
UNSW policies, penalties, and information to help you avoid plagiarism see:
https://student.unsw.edu.au/plagiarism as well as the guidelines in the online ELISE
tutorials for all new UNSW students: http://subjectguides.library.unsw.edu.au/elise .
To see if you understand plagiarism, do this short quiz:
https://student.unsw.edu.au/plagiarism-quiz
The fish.dta dataset contains information on the price and quantity of fish sold at various fish
markets in different locations.
We are interested in the determinants of fish price and we consider the following population
regression models:
????? = ?0 + ?1 ?????? + ?2 ???? + ?,
log⁡(?????) = ?0 + ?1 log⁡(??????) + ?,
(1)
(2)
log(?????) = ?0 + ?1 log(??????) + ?2 ???? + ?,
(3)
????? = ?0 + ?1 ⁡?????? + ???? ?2 + ?,
(4)
log⁡(?????) = ?0 + ?1 log⁡(??????) + ?2 ???? + ?3 ???? 2 + ⁡?,
1/????? = ?0 + ?1 ?????? + ?2 ???? + ⁡ ?3 ?????? ∗ ???? + ?,
(5)
(6)
log(????? + ?0 ) = ?1 ?????? + ?,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ (7)
where ????? is fish price per kg ($100 per kg), totqty is the total quantity sold (kg), wave is the
max height of waves in the last 2 days (meter).
1. [10pts] Which of the above models are linear regression models?
2. [10pts] Using the data in fish.dta, estimate model (1) by OLS and report the results in
equation or tabular form, including the sample size and ? 2 . Assume that the GaussMarkov assumptions hold and interpret the estimated coefficient for ?1 . (Please pay
attention to the unit of measurement when interpreting the coefficient).
3. [8pts] Explain the meaning of the zero-conditional-mean assumption for the model in
equation (1). Interpret ?̂1 if the “zero conditional mean assumption” did not hold.
4. [10pts] Using the data in fish.dta, estimate model (2) by OLS and report the results in
equation or tabular form, including the sample size and ? 2 . Interpret the estimated
coefficients for ?1. (For your convenience I already generated log(price) (lprice) and
log(totqty) (ltotqty). Log here refers to the natural logarithm).
5. [20pts] Consider an extended model of (2) given by (3). For this problem, assume that
the Gauss-Markov assumptions hold for (3).
a. Estimate equation (3) and present the results in equation or tabular form,
including the sample size and ? 2 .
b. Interpret ?̂1 and ?̂2 .
c. Compare ?̂1 and ?̂1 obtained in question 4 (from model (2)). What do you
conclude about the sign of the bias on ?̂1 in model (2)?
d. Given the estimates of equation (3) and the sign of the bias on ?̂1, are log(totqty)
and Wave positively or negatively correlated?
e. Looking at the estimated ?̂2 , can the result be explained by the mechanism that
stormy seas decrease the supply of fish caught and therefore increase the price
of fish?
6. [10pts] Using model (3) and assuming that the Gauss-Markov assumptions hold, is
log(??????)⁡statistically significant at the 1% significance level? Conduct the test
manually and present your conclusion.
7. [8pts] Using the estimates of equation (3) answer the following question. Find the
predicted selling price of a market where 200 kg of fish were sold and the max height
of waves in the last 2 days was 5 meters. You can assume that the Gauss-Markov
assumptions hold.
8. [10pts] Estimate model (5) and use the estimates of equation (5) to answer the following
question. On Saturday, 200 kg of fish were sold in the Sydney fish market and the max
height of waves in the last 2 days was 5 meters. Predict the change or percent change
in fish price in the Sydney fish market if the same amount of fish (200 kg) were sold
but the max height of waves in the last 2 days increased by 1 meter. You can assume
that the Gauss-Markov assumptions hold. (For your convenience I already generated
???? 2 (wavesq)).
9. [6pts] Interpret the ? 2 of the OLS estimates of equations (2) and (3). Explain why the
? 2 is higher for equation (3) than for equation (2).
10. [8pts] We were able to obtain the quantity of fish sold to men and women separately.
Your friend thought that the selling price might depends on the quantity sold to each
group differently, and proposed the following regression to examine the determinants
of fish price.
log(?????) = ?0 + ?1 log(??????) + γ2 log(??????? ) + ?3 log(??????? ) + ?4 ???? + ?
(8)
where ??????? ⁡and ??????? are the total quantities sold to male and female customers,
respectively.
Data were collected in developed countries. Which Gauss-Markov assumptions are
potentially violated in equation (8)? Include all that apply.
Introductory Econometrics
Problem Set 1
Assignment instructions
•
•
•
•
•
•
•
You must submit your work via the Turnitin link on moodle by 18:00 on Friday
March 11.
This assignment will be marked for the course assessment and will be worth 7.5% of
your final mark. You must attach your Stata do-file to your answers (by typing
Stata codes or pasting screenshots of the do-file window or the result window in
your answer) and failure to do so will result in a zero mark for the computing
questions.
Name, student number, course title, tutorial group number and tutor’s name should be
clearly included in the submission. Your answers including Stata do-file should not
exceed 6 pages. The Assignment is based on the material covered in both lectures and
tutorials up to the end of Week 3.
The assignment is INDIVIDUAL work. You may discuss the assignment with your
peers, but you must submit YOUR OWN answers.
If the answer requires some mathematical calculation, show the steps, don’t just report
the final results.
This assignment has a total of 100 points awarded.
All submissions may be checked for plagiarism. The University regards plagiarism as
a form of academic misconduct and has very strict rules regarding plagiarism. For
UNSW policies, penalties, and information to help you avoid plagiarism see:
https://student.unsw.edu.au/plagiarism as well as the guidelines in the online ELISE
tutorials for all new UNSW students: http://subjectguides.library.unsw.edu.au/elise .
To see if you understand plagiarism, do this short quiz:
https://student.unsw.edu.au/plagiarism-quiz
The fish.dta dataset contains information on the price and quantity of fish sold at various fish
markets in different locations.
We are interested in the determinants of fish price and we consider the following population
regression models:
????? = ?0 + ?1 ?????? + ?2 ???? + ?,
log⁡(?????) = ?0 + ?1 log⁡(??????) + ?,
(1)
(2)
log(?????) = ?0 + ?1 log(??????) + ?2 ???? + ?,
(3)
????? = ?0 + ?1 ⁡?????? + ???? ?2 + ?,
(4)
log⁡(?????) = ?0 + ?1 log⁡(??????) + ?2 ???? + ?3 ???? 2 + ⁡?,
1/????? = ?0 + ?1 ?????? + ?2 ???? + ⁡ ?3 ?????? ∗ ???? + ?,
(5)
(6)
log(????? + ?0 ) = ?1 ?????? + ?,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ (7)
where ????? is fish price per kg ($100 per kg), totqty is the total quantity sold (kg), wave is the
max height of waves in the last 2 days (meter).
1. [10pts] Which of the above models are linear regression models?
2. [10pts] Using the data in fish.dta, estimate model (1) by OLS and report the results in
equation or tabular form, including the sample size and ? 2 . Assume that the GaussMarkov assumptions hold and interpret the estimated coefficient for ?1 . (Please pay
attention to the unit of measurement when interpreting the coefficient).
3. [8pts] Explain the meaning of the zero-conditional-mean assumption for the model in
equation (1). Interpret ?̂1 if the “zero conditional mean assumption” did not hold.
4. [10pts] Using the data in fish.dta, estimate model (2) by OLS and report the results in
equation or tabular form, including the sample size and ? 2 . Interpret the estimated
coefficients for ?1. (For your convenience I already generated log(price) (lprice) and
log(totqty) (ltotqty). Log here refers to the natural logarithm).
5. [20pts] Consider an extended model of (2) given by (3). For this problem, assume that
the Gauss-Markov assumptions hold for (3).
a. Estimate equation (3) and present the results in equation or tabular form,
including the sample size and ? 2 .
b. Interpret ?̂1 and ?̂2 .
c. Compare ?̂1 and ?̂1 obtained in question 4 (from model (2)). What do you
conclude about the sign of the bias on ?̂1 in model (2)?
d. Given the estimates of equation (3) and the sign of the bias on ?̂1, are log(totqty)
and Wave positively or negatively correlated?
e. Looking at the estimated ?̂2 , can the result be explained by the mechanism that
stormy seas decrease the supply of fish caught and therefore increase the price
of fish?
6. [10pts] Using model (3) and assuming that the Gauss-Markov assumptions hold, is
log(??????)⁡statistically significant at the 1% significance level? Conduct the test
manually and present your conclusion.
7. [8pts] Using the estimates of equation (3) answer the following question. Find the
predicted selling price of a market where 200 kg of fish were sold and the max height
of waves in the last 2 days was 5 meters. You can assume that the Gauss-Markov
assumptions hold.
8. [10pts] Estimate model (5) and use the estimates of equation (5) to answer the following
question. On Saturday, 200 kg of fish were sold in the Sydney fish market and the max
height of waves in the last 2 days was 5 meters. Predict the change or percent change
in fish price in the Sydney fish market if the same amount of fish (200 kg) were sold
but the max height of waves in the last 2 days increased by 1 meter. You can assume
that the Gauss-Markov assumptions hold. (For your convenience I already generated
???? 2 (wavesq)).
9. [6pts] Interpret the ? 2 of the OLS estimates of equations (2) and (3). Explain why the
? 2 is higher for equation (3) than for equation (2).
10. [8pts] We were able to obtain the quantity of fish sold to men and women separately.
Your friend thought that the selling price might depends on the quantity sold to each
group differently, and proposed the following regression to examine the determinants
of fish price.
log(?????) = ?0 + ?1 log(??????) + γ2 log(??????? ) + ?3 log(??????? ) + ?4 ???? + ?
(8)
where ??????? ⁡and ??????? are the total quantities sold to male and female customers,
respectively.
Data were collected in developed countries. Which Gauss-Markov assumptions are
potentially violated in equation (8)? Include all that apply.
2 / 3
125%
=
log(price + B.) = Bitotqty + R,
(7)
where price is fish price per kg ($100 per kg), totqty is the total quantity sold (kg), wave is the
max height of waves in the last 2 days (meter).
1. [10pts] Which of the above models are linear regression models?
2. [10pts] Using the data in fish.dta, estimate model (1) by OLS and report the results in
equation or tabular form, including the sample size and R2. Assume that the Gauss-
Markov assumptions hold and interpret the estimated coefficient for Q1. (Please pay
attention to the unit of measurement when interpreting the coefficient).
3. [8pts] Explain the meaning of the zero-conditional-
mean assumption for the model in
equation (1). Interpret âı if the “zero conditional mean assumption” did not hold.
4. [10pts] Using the data in fish.dta, estimate model (2) by OLS and report the results in
equation or tabular form, including the sample size and R2. Interpret the estimated
coefficients for $1. (For your convenience I already generated log(price) (Iprice) and
log(totqty) (ltotqty). Log here refers to the natural logarithm).
5. [20pts] Consider an extended model of (2) given by (3). For this problem, assume that
the Gauss-Markov assumptions hold for (3).
a. Estimate equation (3) and present the results in equation or tabular form,
including the sample size and R2.
b. Interpret Û1 and Û2 .
c. Compare ûn and ß1 obtained in question 4 (from model (2)). What do you
conclude about the sign of the bias on ß, in model (2)?