THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF ECONOMICS ECON1203/ECON2292 (ARTS) QUANTITATIVE METHODS B FINAL EXAMINATION Session1, 2006 Time allowed: Three hours. Total marks: 65 marks. There are FIVE questions in this examination. Answer ALL five questions. The marks assigned to each question are NOT of equal value. The value of each sub-question is indicated in brackets. On the front of your answer book, write the number of each question you have attempted. Statistical tables and useful formulae are attached to this examination paper. Electronic calculators may be used. The examination paper may be retained by the candidate. Answers must be written in ink. Pencils may be used only for drawing, sketching or graphical work. Show the working steps in your answers.
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THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF ECONOMICS
ECON1203/ECON2292 (ARTS) QUANTITATIVE METHODS B
FINAL EXAMINATION
Session1, 2006
Time allowed: Three hours.
Total marks: 65 marks.
There are FIVE questions in this examination.
Answer ALL five questions.
The marks assigned to each question are NOT of equal value.
The value of each sub-question is indicated in brackets.
On the front of your answer book, write the number of each question you have attempted.
Statistical tables and useful formulae are attached to this examination paper.
Electronic calculators may be used.
The examination paper may be retained by the candidate.
Answers must be written in ink. Pencils may be used only for drawing, sketching orgraphical work.
Show the working steps in your answers.
Question 1 [13 marks in total]
(a) Suppose that the annual percentage change in advertisement spending for arandomly-selected company listed in a large stock exchange for the year 2001approximately follows a normal distribution. The mean and standard deviation ofa random sample of changes in ad spending for 16 companies turn out to be 5%and 12% respectively.
(i) Assume that the population standard deviation is known to be 12%. Aconfidence interval for the mean change in ad spending is constructed. Itsupper confidence limit is 9.935%. What is the level of confidence?[2 marks]
(ii) Assume that the population standard deviation is unknown. Find the 95%confidence interval for the mean change in ad spending. [3 marks]
(b) A financial adviser uses the variance of returns from the set of stocks sherecommends as a measure of investment risk. The set consists of many stocks andthe distribution of the stock annual returns is approximately normal. She has arandom sample with 15 stock returns from the recommended set. The observedsample variance is 0.0256.
(i) Construct the 90% confidence interval for the population variance of thereturns. [2 marks]
(ii) Interpret the numerical result in (i) in terms of the definition of the 90%confidence interval. [2 marks]
(c) To decide whether or not a soccer game series should be aired, a commercial TVchannel needs to estimate the proportion of households that are interested in thesoccer series. Suppose that the TV channel's management wishes to have a 95%confidence interval for the proportion, of which the width is 0.06 (i.e. B = 0.03).
(i) Suppose that you are able to interview households about whether or notthey will watch the soccer series. Outline how would you construct therequired confidence interval? [2 marks]
(ii) Determine the minimum sample size that satisfies the above requirement.[2 marks]
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Question 2 [10 marks in total]
(a) A watch running more than one minute faster or slower than the standard time ina year is regarded as defective. QWatches, a digital watch manufacturer, claimsthat the probability of its watch being defective is 0.05. Having received somecomplaints, a consumer protection agency plans to conduct a survey about thewatches made by QWatches.
(i) If QWatches' claim is correct, what is the probability that more than onewatch out of 10 surveyed consumers are defective? [1 mark]
(ii) If QWatches' claim is correct, use an appropriate approximation to find theprobability that more than 3 watches out of 80 surveyed consumers aredefective. [2 marks]
(iii) If QWatches' claim is correct, use an appropriate approximation to find theprobability that more than 8 watches out of 160 surveyed consumers aredefective. [2 marks]
(b) The City of Wildwood issues fine tickets to individuals who park in no-parkingzones. The fine depends on the day of week and the time of day. Historical dataindicate that 40%, 30%, 20% and 10% of the fine tickets are respectively for $20,$40, $60 and $80. Let Xbe the fine on a randomly-selected fine ticket. Find theexpected value and the standard deviation ofX. [2 marks]
(c) Anna, Brad and Candy work for the billing division of an electricity supplier,respectively handling 30%, 50% and 20% of the monthly bills. Historical datashow that 1%,2% and 3% of the bills handled respectively by Anna, Brad andCandy contain errors. Suppose that the division receives a complaint about abilling error. What is the probability that the error is made by Brad? [3 marks]
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Question 3 [12 marks in total]
(a) Define what is meant by an unbiased estimator. Give two examples of estimatorsthat are unbiased. [2 marks]
(b) State the Central Limit Theorem. [2 marks]
(c) The number of hits received by a web-page during the hour 9am-l Oam varies from oneday to another. Below is a sample of the hits (already ranked) for the hour over 16 days.
25
36
36
47
47
58
59
517
(i) Find the median and the mode of this data set. [2 marks]
(ii) Find the first and third quartiles of this data set. [2 marks]
(d) In a corporation, a very small number of employees have extremely high salaries,whereas the majority of employees receive much lower salaries. Ifyou were thebargaining agent for the union, what statistic would you use to illustrate that theemployees have a low level of salaries? Why? [2 marks]
(e) A sample of marks from 12 students for a test in an economics subject yields amean of 25 and a standard deviation of 4. Suppose that the sample is enlarged to14 data points by including two additional marks with a common value of25.Find the mean and standard deviation of the enlarged sample. [2 marks]
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Question 4 [15 marks in total]
(a) In hypothesis testing, data are used to examine a null hypothesis about a populationparameter against an alternative hypothesis. In deciding whether or not the nullhypothesis should be rejected,
(i) what is a Type I error and what is a Type II error? [I mark](ii) why is the level of significance usually chosen as a small positive
number? [I mark]
(b) People have different views on a reform for the national health care system. A surveyconsisting of 200 adults has been carried out. The survey data are summarised in the tablebelow. David is interested in the hypothesis that a person's view on the reform isindependent of the person's gender. By using the technique of hypothesis testing (X2 test),he found that the hypothesis could not be rejected. He used the 5% level of significancein his test.
Views on the reformGender For Against NeutralFemale 58 30 15Male 41 40 16
Repeat David's test to check ifhis conclusion is correct. Write down thehypotheses, the test statistic, the decision rule and your conclusion. [5 marks]
(c) Fund managers aim to achieve high returns for their funds. A newspaper article claimsthe mean return achieved by the fund managers in 2003 is 5%. A random sampleconsisting of the returns in 2003 from 16 fund managers indicates that the mean is -I %and the standard deviation is 8%. Test the null hypothesis that the mean return in 2003 is5% against the alternative hypothesis that the mean return in 2003 is less than 5%,assuming that the return is approximately normally distributed. Use the 5% level ofsignificance. [3 marks]
(d) A government spokesman claims that 55% of voters support a recent tax cut.Sam, an accounting graduate, decides to test the spokesman's claim by using thefollowing procedure. First, 10 independent voters will be interviewed about thetax cut. Second, the number of voters supporting the tax cut will be recorded.Finally, the claim should be rejected if the number of supporters is less than 4.
(i) Write down the null and alternative hypotheses Sam are interested in.[1 mark]
(ii) Find the level of significance ofSam's test. [2 marks](iii) Find the power of Sam's test when the proportion of supporters is 45%.
[2 marks]
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Question 5 [15 marks in total]
(a) For the simple linear regression model r; = Po + PjX; + 8; to be valid, the error
term c; must satisfy certain conditions. What are these conditions? [2 marks]
(b) The following summary output is from a regression equation estimated usingMicrosoft Excel. The dependent variable (Y) is the interest rate on a short-termgovernment bond and the independent variable (X) is the inflation rate. Both theinterest rate and the inflation rate are expressed as percent per year. The model is
r; = Po + PjX; + 8; .
Quarterly data from March 2000 to March 2005 comprising twenty-oneobservations (i.e.,n = 21) were used to estimate the regression equation.
Regression StatisticsR SquareStandard Error of the RegressionObservations
0.27500.5444
21
InterceptX Variable
Coefficients Standard Error4.4602 0.32190.2408 0.0897
t Stat13.85752.6845
P-value0.00000.0147
(i) Find the 95% confidence interval for PI. [2 marks](ii) Test the null hypothesis that PI = 0 against the alternative that P, > 0 at the
5% level of significance. Write down your decision rule and conclusion.[2 marks]
(iii) Suppose that the inflation rate (X) is 5%. Make a point "prediction" for theinterest rate (Y) by using the estimated linear regression model. [2 marks]
(iv) The "R Square" or "coefficient of determination" is 0.275. Explain themeaning of"R Square". [2 mark]
(v) The p-value "0.0147" in the above table is a probability. Exactly, what isthis probability? [2 marks]
(c) To find the relationship between a company's expenditure on advertisements (X)and sales (Y), an analyst collected 6 Quarterly observations on X and Y of thecompany. The means and sums of squares for the two variables from the sampleare computed:
Test the null hypothesis that X and Y are uncorrelated against the alternative thatthey are positively correlated, using the 5% level of significance. Write downyour decision rule and conclusion. [3 marks]