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ECO252 QBA2
FIRST EXAM
October 4 and 8, 2007
Version 2
Name: ________________
Class hour: _____________
Student number: __________
Show your work! Make Diagrams! Include a vertical line in the middle! Exam is normed on 50
points. Answers without reasons are not usually acceptable.
I. (8 points) Do all the following.
x ~ N 3,13 
1. Px  0
2. P 40  x  2
3. P 3  x  3
4. x.125 (Do not try to use the t table to get this.)
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II. (9 points-2 point penalty for not trying part a.)
Our sales of microwave ovens in five randomly picked months appear below
122 126 140 142 150
a. Compute the sample standard deviation, s , of expenditures. Show your work! (2)
b. Assuming that the underlying distribution is Normal, compute a 99% confidence interval for the
mean. (2)
c. Redo b) when you find out that there were only 14 months to pick the data from.(2)
d. Assume that the population standard deviation is 10 and create a 75% two-sided confidence
interval for the mean. (2)
e. Use your results in a) to test the hypothesis that the mean is below 140 at the 99% level. (3)
State your hypotheses clearly!
f. (Extra Credit) Given the data, test the hypothesis that the population standard deviation is below
15 (3)?
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III. Do as many of the following problems as you can.(2 points each unless marked otherwise adding to
13+ points). Show your work except in multiple choice questions. (Actually – it doesn’t hurt there
either.) If the answer is ‘None of the above,’ put in the correct answer if possible.
1) If I want to test to see if the mean of x is smaller than the given population mean  0 my null
hypothesis is:
i)    0
ii)    0
iii)    0
iv)    0
v) Could be any of the above. We need more information.
vi) None of the above
2) Assuming that you have a sample mean of 100 based on a sample of 36 taken from a population of 300
with a known population standard deviation of 80, the 99% confidence interval for the population mean is
 80 

a) 100  2.724 
 300 
 300  36 80 

b) 100  2.724 
 300  1 36 


 80 

c) 100  2.724 
 36 
 80 

d) 100  2.438 
 300 
 300  36 80 

e) 100  2.438 
 300  1 36 


 80 

f) 100  2.438 
 36 
 300  36
g) 100  2.438 
 300  1

80 
300 
 80 

h) 100  2.576 
 36 
 300  36 80 

i) 100  2.576 
 300  1 36 


 80 

j) 100  2.576 
 300 
k) None of the above. Fill in a correct answer.
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3) Which of the following is a Type 2 error?
a) Rejecting the null hypothesis when the null hypothesis is false.
b) Not rejecting the null hypothesis when the null hypothesis is false.
c) Rejecting the null hypothesis when the null hypothesis is true.
d) Not rejecting the null hypothesis when the null hypothesis is true.
e) All of the above
f) None of the above.
4) If a random sample is gathered to get information about a population proportion, what do we mean by a
p-value?
a) P-value is the population proportion in the null hypothesis.
b) P-value is the population proportion in the alternate hypothesis.
c) P-value is the probability that, if the null hypothesis was false, that, if we were to repeat the
experiment many times, we would get a sample proportion as extreme as or more extreme
than the sample proportion actually observed.
d) P-value is the probability that, if the null hypothesis was true, that, if we were to repeat the
experiment many times, we would get a sample proportion as extreme as or more extreme
than the sample proportion actually observed.
e) P-value is the probability that the alternate hypothesis is true, given the sample proportion
actually observed.
f) P-value is the probability of a type 2 error.
g) None of the above is true.
5) If a difference in proportions (in a business-related problem) is called statistically significant at the 1%
significance level, this means that
a) If the null hypothesis is true, the difference in proportions is surprisingly small.
b) We must reject the null hypothesis.
c) The difference in proportions is small enough so that we must take account of it in our
business decisions.
d) The null hypothesis is very likely to be true.
e) All of the above
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Assume a Normal distribution in 6) and 7).
6) (Wonnacott & Wonnacott) A company is discharging treated waste into a river. The firm is supposed to
be fined if the average pollution level is above 16 parts per million. It is known that the population standard
deviation is 6 parts per million. 9 measurements are taken. If we assume that the firm is discharging 16
parts or fewer per million, how high must the sample mean level of pollution be to cause us to doubt the
assumption? (Use a 1% significance level, and don’t forget to state your hypotheses.) (3)
7) The politicians, who don’t know any statistics, decide that they will fine the company in 6) if the level of
pollution exceeds 19 parts per million. . It is known that the population standard deviation is 6 parts per
million. 9 measurements are taken. If we assume that the firm is discharging 16 parts per million what is
the probability that they will be fined? (Think p-value?) (2)
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8) It is a well-known fact that your factory has been producing a product that is 20% defective. We take a
sample of 500 units of the product this month and find that 103 are defective.
a) Assuming that the 20% figure is correct, how many units of the product must be examined
before we can state our defect rate as a proportion .03 ? (Use a 99% confidence level)
b) If we wish to test our belief that the proportion has risen over the previous figure, let p
represent the proportion of defective items. What are our null and alternative hypotheses? (1)
c) You already know that the sample size is 500. Using a 99% confidence level and assuming that
your hypothesis in b) is correct, test the hypothesis. (2)
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Blank page for calculations.
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ECO252 QBA2
FIRST EXAM
October 8, 2007
TAKE HOME SECTION
Name: _________________________
Student Number and class: _________________________
IV. Do at least 3 problems (at least 7 each) (or do sections adding to at least 20 points - Anything extra
you do helps, and grades wrap around) . Show your work! State H 0 and H 1 where appropriate. You
have not done a hypothesis test unless you have stated your hypotheses, run the numbers and stated
your conclusion. (Use a 95% confidence level unless another level is specified.) Answers without
reasons usually are not acceptable. Neatness and clarity of explanation are expected. This must be
turned in when you take the in-class exam. Note that answers without reasons and citation of
appropriate statistical tests receive no credit. Failing to be transparent about which section of which
problem you are doing can lose you credit. Many answers require a statistical test, that is, stating or
implying a hypothesis and showing why it is true or false by citing a table value or a p-value. If you haven’t
done it lately, take a fast look at ECO 252 - Things That You Should Never Do on a Statistics Exam (or
Anywhere Else).
A group of 30 employees are interviewed to determine the minimum amount that they will take to give up a
vacation day. After careful interviewing, a psychologist repots the following amounts.
479 648 522 595 547 657 578 539 553 520 499 606 612
616 488 557 621 628 511 634 625 612 509 633 616 598
627 622 512 631
x  17395 and that the sum of squares is
My calculations say that the sum of these 30 numbers is
x

2
 10171575 . This is a sample of 30.
Personalize these data as follows. Take the second to last digit of your student number and multiply it by 5.
Add this quantity to each of the 30 numbers. If the second to last digit of your student number is 0, add 50.
Label your exam by version number as follows. If the second to last digit of your student number is 1, you
are doing Version 1. If the second to last digit is 2, you are doing Version 2 etc. If the second to last digit is
zero, you are doing version 10. Last term's exam said the following.
If you add a quantity a to a column of numbers,
 x  a   x na,
 x  a    x  2a x na . For example, if a  60 ,
 x  60    x 3060 ,  17395 + 1800 = ? and
 x  60   x  260 x 3060  1017157512017395  303600.
2
2
2
2
2
2
Test the following
Problem 1: Count the number of people in your sample that demand more than $602.50 and make it into a
sample proportion. Test the following 3 hypotheses: I) that 60% demand more than $602.50, II) that more
than 60% demand more than $602.50 and III) that less than 60% demand more than $602.50, using a 98%
confidence level.
For each of these three tests a) state your null and alternative hypotheses (2), b) test each one using
a test ratio or a critical value for the proportion (2) and c) find a p-value for the null hypotheses (3). Label
each part clearly so that I know which is I, II and III and a), b) c). Make sure that I know where the ‘reject’
zone is.
d) Using the proportion you found above, how large a sample would you need to estimate a 2-sided 98%
confidence interval for the proportion with and error of at most .001? Assume that your sample is of that
size and show that the confidence interval has an error of at most .001. (3)
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e) (Extra credit) Assume that you are testing the hypothesis that (II) more than 60% demand over $602.50,
find the power of the test if you use a sample of 30 the true proportion is 70% (3)
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Problem 2: Assume that the underlying data for problem 1 is not Normal and using the data for problem 1
test the following three hypotheses: I) that the median demand is $602.50, II) that median demand is more
than $602.50 and III) that the median demand is less than $602.50, using a 98% confidence level. a) state
your null and alternative hypotheses and the hypotheses that you will actually test for each of the 3 tests
(3), b) test each one using a test ratio or a critical value (3), c) find a p-value for the 2-sided test and explain
whether and why it would lead to a rejection of the null hypothesis at the 95% confidence level (1), d)
(extra credit) Show explicitly what the conclusion in c) would be if the sample of 30 came from a
population of 60. (1) e) (extra credit) find a two sided confidence interval for the median (2)
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Problem 3: a) Find the sample mean and sample standard deviation for the data in Problem 1 (1)
b) Test the hypothesis that the mean is 602.50 using critical values for the sample mean, first stating your
hypotheses clearly. Use a 98% confidence level (2)
c) Test the hypothesis in b) using a test ratio. Find an approximate p-value and state and explain whether
this will lead to a rejection of the null hypothesis if we continue to use a 98% confidence level. (2)
d) Using the test ratio you found in c) find a p-value for the null hypothesis that the mean is at most 602.50
(1)
e) Using the test ratio you found in c) find a p-value for the null hypothesis that the mean is at least 602.50
(1)
f) Test the null hypothesis that the mean is at most 602.50 using an appropriate confidence interval (1)
g) Test the null hypothesis that the mean is at least 602.50 using an appropriate confidence interval (1)
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Problem 4: Assume that the population standard deviation is known to be 30 but that we are still working
with a problem like Problem 3. (98% confidence level, sample of 30.) Do either Problem 4.1 or Problem
4.2. Make sure that I know which one!
Problem 4.1. a) Find a critical value for the sample mean if we are testing whether the population
mean is below 30. Clearly state your null and alternative hypotheses (2)
b) Assume that the sample mean is 30 minus the second to last digit of your student number. (Use 10 if this
digit is zero.) Find a p-value for your null hypothesis. (1)
c) Create a power curve for the test (6)
Problem 4.2. a) Find critical values for the sample mean if we are testing whether the population
mean is 30. Clearly state your null and alternative hypotheses (2)
b) Assume that the sample mean is 30 minus the second to last digit of your student number. (Use 10 if this
digit is zero.) Find a p-value for your null hypothesis. (1)
c) Create a power curve for the test (8)
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Problem 5: In problem 4 we assumed that the population standard deviation is 30.
a) Do a 98% confidence interval for the mean using the mean that you found in Problem 3 and assuming
that our sample of 30 came from a population of 300. (2)
b) How large a sample would we need if we wanted to make the error term no more than 1 and the
sample came from an infinite population? (2)
c) Using a 98% confidence level and a sample size of 30 create a confidence interval for the population
standard deviation using your sample variance or standard deviation from Problem 3. (2)
d) Repeat c) assuming that you had a sample of 300. (2)
e) Can we say that the standard deviation is significantly different from 30 on the basis of c) and d)? (1)
f) Using the data and sample size from problem 3 can we say that the standard deviation is above 30? State
your hypotheses and do an appropriate hypothesis test. (3)
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