252 1takehome072 10/3/07
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 reports the following amounts.
479
616
627
648
488
622
522
557
512
595
621
631
547
628
657
511
578
634
539
625
My calculations say that the sum of these 30 numbers is
x
2
553
612
520
509
499
633
606
616
612
598
 x  17395 and that the sum of squares is
 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 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 an 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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252 1takehome072 10/3/07
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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