252x0753 9/26/07 (Open in ‘Print Layout’ format)
ECO252 QBA2
FIRST EXAM
October 4 and 8, 2007
Version 3
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 4, 11
1. Px  0
2. P33  x  2
3. P4  x  4
4. x.075 (Do not try to use the t table to get this.)
1
252x0753 9/26/07 (Open in ‘Print Layout’ format)
II. (9 points-2 point penalty for not trying part a.)
Monthly incomes (in thousands) of 6 randomly picked individuals in the little town of Rough Corners are
shown below.
2.5 7.3 3.1 2.6 2.4 3.0
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 50 people living in Rough Corners. (2)
d. Assume that the population standard deviation is 2 and create an 85% two-sided confidence
interval for the mean. (2)
e. Use your results in a) to test the hypothesis that the mean income is above 2.3(thousand) 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
2. (3)
2
252x0753 9/26/07 (Open in ‘Print Layout’ format)
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 sample standard deviation of 80, the 99% confidence interval for the population mean is
 80 

a) 100  2.576 
 36 
 300  36 80 

b) 100  2.576 
 300  1 36 


 80 

c) 100  2.576 
 300 
 80 

d) 100  2.724 
 300 
 300  36 80 

e) 100  2.724 
 300  1 36 


 80 

f) 100  2.724 
 36 
 80 

g) 100  2.438 
 300 
 300  36 80 

h) 100  2.438 
 300  1 36 


 80 

i) 100  2.438 
 36 
 300  36
j) 100  2.438 
 300  1

80 
300 
g) None of the above. Fill in a correct answer.
3
252x0753 9/26/07 (Open in ‘Print Layout’ format)
3) Which of the following is a Type 2 error?
a) Rejecting the null hypothesis when the null hypothesis is true.
b) Not rejecting the null hypothesis when the null hypothesis is true.
c) Not rejecting the null hypothesis when the null hypothesis is false.
d) Rejecting the null hypothesis when the null hypothesis is false.
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 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.
b) 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.
c) P-value is the population proportion in the null hypothesis.
d) P-value is the population proportion in the alternate hypothesis.
e) P-value is the probability of a type 2 error.
f) P-value is the probability that the alternate hypothesis is true, given the sample proportion
actually observed.
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 large.
b) There is a 99% chance that the null hypothesis is true.
c) The difference in proportions is large enough so that we must take account of it in our
business decisions.
d) All of the above
6) (Wonnacott & Wonnacott) When an industrial process is in control, it produces bolts with a hardness
that has a mean of at least 80 and a (population) standard deviation of 8. If the hardness is too far below 80,
you must shut down the process. Every hour you take a sample of 16 bolts. How low must the average
hardness of these bolts be before we shut the process down? (Use a 10% significance level, and don’t forget
to state your hypotheses) (3)
7) Your boss, who doesn’t know any statistics, tells you to shut down the process in 6) if the hardness level
from a sample of 16 bolts is 79 or lower. . It is known that the population standard deviation is 8. If we
assume that the process is producing bolts with an average hardness of 80, what is the probability that it
will be shut down? (Think p-value?) (2)
[15]
4
252x0753 9/26/07 (Open in ‘Print Layout’ format)
8) A 1989 Gallup poll revealed that 59% of women believed that the Republican Party were more likely
than the Democratic Party to keep the country prosperous (My, how things change!). We were already sure
that 68% of men believed that this was true.
a) How many women had to be polled before we could state that the proportion for women is
.59  .03 ? (Use a 5% significance level.) (2)
b) If we wished to test our belief that women were less likely to believe that the Republicans were
more likely than the Democratic Party to keep the country prosperous and we were already sure that 68%
of men believed that this was true, let p represent the proportion of women. What are our null and
alternative hypotheses? (1)
c) The actual poll covered a sample of 750 women. Using a 95% confidence level and assuming
that your hypothesis in b) is correct, test the hypothesis. (2)
[20]
5
252x0753 9/26/07 (Open in ‘Print Layout’ format)
Blank page for calculations
6
252x0753 9/26/07 (Open in ‘Print Layout’ format)
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
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 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)
[10]
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)
7
252x0753 9/26/07 (Open in ‘Print Layout’ format)
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)
[17]
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)
[26]
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)
[37]
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)
[49]
8