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ECO252 QBA2
FIRST HOUR EXAM
October 17-18 2005
Version 1
Name ________________
Hour of class registered _____
Class attended if different ____
Show your work! Make Diagrams! Exam is normed on 50 points. Answers without reasons are not
usually acceptable.
I. (8 points) Do all the following.
x ~ N 6, 6.5
1. P 17.5  x  0
2. P6.00  x  7.00 
3. Px  10.22 
4. x.03 (Do not try to use the t table to get this.)
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II. (5 points-2 point penalty for not trying part a.) In order not to violate the truth in labeling law a teabag
must contain at least 5.5 oz of tea. A sample of 9 items is taken from a large number of tea bags. The data
below is found. (Recomputing what I’ve done for you is a great way to waste time.) b) and d) require
statistical tests.
a. Compute the sample standard deviation, s , of the waiting times. Show your work! (2)
b. Is the population mean significantly below 5.5 (Use a 95% confidence level)? Show your
work! (3)
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c. (Extra Credit) Find an approximate p value for your null hypothesis. (2)
d. Assume that the population standard deviation is 0.10 and create a 94% confidence interval for
the mean. (2)
e. (Extra Credit) Given the data, is 0.10 a reasonable value for the population standard deviation?
x
Row
x2
1
2
3
4
5
6
7
8
9
5.36 28.7296
5.17 26.7289
5.40 29.1600
5.38 28.9444
5.46 29.8116
5.50 30.2500
5.62 31.5844
5.35 28.6225
5.42 29.3764
48.66 263.2078
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III. Do as many of the following problems as you can.(2 points each unless marked otherwise adding to
18+ 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.
1.
Which of the following is a Type 2 error?
a) Rejecting the null hypothesis when the null hypothesis is false.
b) Rejecting the null hypothesis when the null hypothesis is true.
c) Not rejecting the null hypothesis when the null hypothesis is true.
d) Not rejecting the null hypothesis when the null hypothesis is false.
e) All of the above
f) None of the above.
2.
A librarian provides a confidence interval estimate for the mean number of books checked out
daily. The estimate is 229 to 741. The point estimate that this interval is based on is.
a) 229.
b) 485
c) 741
d) 970
e) None of the above
f) There is not enough information to tell.
3.
(BLK8.30) If we want a 95% confidence interval for the average income of in a town, and the
population standard deviation is known to be $1000. We have taken a sample of size 50 earlier and
found that the sample mean is $14800. What sample size should we take if the width of the
interval is to be no more than $100?
a) 1537
b) 385
c) 50
d) 40
e) 20
f) None of the above.
4.
An entrepreneur is considering the purchase of a coin-operated laundry. The present owned claims
that over the past 6 years the average daily income was $700. A sample is taken of daily revenue
over a period of 28 days. Statistics are computed from the sample. If we want to test the statement
that the mean is $700, which of the following tests is most appropriate? (1 point)
a) z -test of a population mean.
b) z -test of a population proportion.
c) t -test of a population mean.
d)  2 -test of a population variance.
e) F -test.
f) All of the above could be used.
g) We do not have enough information.
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5.
We are trying to estimate the median income in a region. We wish to test if the median is over
$30000. We do not know the population variance. We can compute a statistic or statistics from a
sample of 27 incomes. To do this test, the statistic or statistics we need is (are)
a) The number of incomes in the sample that are above $30000.
b) The sample mean, x .
c) The sample mean, x and the sample variance s 2 .
d) The sample median x.50 .
e) The sample variance s 2 .
f) The proportion of incomes that are above the sample mean, x .
g) The proportion of incomes that are above the sample median x.50 .
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6.
A study of child support says that the average support paid by noncustodial fathers is $370/month.
We hope that fathers in our city are paying more than the national average. A random sample of
324 custodial mothers is taken. The results are a sample mean of $385.46 and a sample standard
deviation of $35. Test whether this mean is significantly above $370.   .05 
a) State your null and alternative hypotheses. (1)
b) Test your hypothesis in a) by finding a critical value for the sample mean. Can we say
that the result of the sample is above $370? Why? (3)
c) Do a 2-sided confidence interval for the mean. (2)
d) Do a 2-sided confidence interval for the variance. (2)
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e) (Extra credit) Explain in as much detail as reasonable how you would find a confidence
interval for the median. (2)
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7.
Return to the problem in Question 6. Let’s say that the sample mean payment of $385.46 and the
sample standard deviation of $35 come from a sample of 49. Assume that your null hypothesis is
correct and that you get a t-ratio of 3.091. What should you say that the p-value is? (3 points).
Note that showing your calculations here could get you partial credit.
a) Exactly .001
b) Exactly .002
c) Between .01 and .005
d) Between .02 and .01
e) Between .005 and .001
f) Between .01 and .002
g) Exactly .999
h) Exactly .998
i) Between .99 and .995
j) Between .98 and .99
k) Between .995 and .999
l) None of the above – Show your answer!
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8.
(Mann) According to a 1992 survey, 45% of workers say that they would change careers if they
could. You wish to show that the proportion of workers in your union that want to change careers
is above the national figure. Find the correct set of hypotheses below.
 H : p  .45
a)  0
 H1 : p  .45
b)
 H 0: p  .45

 H1 : p  .45
c)
 H 0: p  .45

 H1 : p  .45
d)
 H 0: p  .45

 H1 : p  .45
e)
 H 0: p  .45

 H1 : p  .45
f)
 H 0: p  .45

 H1 : p  .45
 H 0: p  .45

 H1 : p  .45
h) None of the above. Put in your answer!
g)
9.
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(Mann) According to a 1992 survey, 45% of workers say that they would change careers if they
could. You wish to show that the proportion of workers in your union that want to change careers
is above average. Assume that your null hypothesis in 8 is correct, that you take a sample
of 350 workers and that you compute the ratio z 
p  .45
. If your confidence level is 90%,
.45.55 
350
you should do the following.
a) Reject the null hypothesis if the ratio is not between -1.96 and 1.96.
b) Reject the null hypothesis if the ratio is not between -1.645 and 1.645
c) Reject the null hypothesis if the ratio is above 1.645
d) Reject the null hypothesis if the ratio below -1.645
e) Reject the null hypothesis if the ratio is below 1.282
f) Reject the null hypothesis if the ratio is above 1.282
g) Reject the null hypothesis if the ratio is below -1.282
h) None of the above. Put in your answer!
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10. We wish to use a 2-sided confidence interval to test a proportion. The following things may
influence the size of a confidence interval: (.5 points for good ones, .5 off for bad ones, at worst
zero)
1. Decreasing x to make it closer to .5n
2. Increasing x from .85 n to .9n
3. Changing the null hypothesis to make p 0 closer to .5
4. Increasing the sample size. (Assume a large population)
5. Decreasing the sample size. . (Assume a large population)
6. Increasing the confidence level
7. Increasing the significance level
8. Using a continuity correction with a relatively small sample.
9. Decreasing the population size from 15 n to 14 n
10. Increasing the population size from 5n to 6 n
Put down the numbers of the things that make the confidence interval larger. _________________
You may use rest of this page for calculations. If you use any other paper for
calculations, turn it in – it can only help you.
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ECO252 QBA2
FIRST EXAM
October 11-12 2006
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.
1. (Moore, Notz) You are thinking that it may be desirable to start a wellness program for your
(large) company. You are told that the company will only start such a program if you can show
that the blood pressure of a group of mid-level executives is above normal. The individuals are all
between 35 and 44 years old and US statistics show that mean systolic blood pressure for men in
that age range is 128. You take a sample of 72 executives and get the following results.
x
139.60
136.21
114.18
128.45
120.68
127.51
128.07
161.43
161.09
130.61
130.15
154.74
126.29
124.48
117.51
138.22
169.05
116.14
182.53
141.96
122.14
163.18
124.61
100.03
130.77
140.35
158.74
120.02
137.23
127.22
141.54
105.67
149.55
109.52
131.40
126.54
118.77
141.15
150.30
126.93
144.71
127.32
136.69
125.06
135.21
149.44
133.89
118.37
124.80
133.00
131.74
135.69
169.61
126.71
107.30
122.73
125.35
152.64
109.62
116.59
132.00
117.84
120.01
117.47
145.25
159.94
112.34
145.10
119.39
127.67
117.97
112.40
To personalize the data below take the last digit of your student number, divide it by 10 and add it to the
numbers below. If the last digit of your student number is zero, add 1.00. Label the problem ‘Version 1,’
‘Version 2,’ … ‘Version 10’ according to the number that you used. (For example, Seymour Butz’s
student number is 976502, so he will add 0.20 and change the data to 139.84, 130.81, 182.73 etc. – but see
the hint below, you do not need to write down all the numbers that you are using, just your computations.)
x  9528 .41
Hint - if you use the computational formula: For the original numbers n  72 ,
and
x

2
 1280763 .7 . If you add a quantity a to a column of numbers,
 x  a   x na,  x  a    x  2a x na
2
2
2
Assume that the Normal distribution applies to the data and use a 99% confidence level.
a. Find the sample mean and sample standard deviation of the incomes in your data, showing
your work. (1) (Your mean should be fairly near 132 and your sample standard deviation should
be near 16 or 17.)
b. State your null and alternative hypotheses (1)
c. Test the hypothesis using a test ratio (1)
d. Test the hypothesis using a critical value for a sample mean. (1)
e. Test the hypothesis using a confidence interval (1)
f. Find an approximate p-value for the null hypothesis. (1)
g. On the basis of your tests, will you get a wellness program? Why? (1)
h. How do your conclusions change if the sample of 72 is taken from a population of 200? (2)
i. Assume that the Normal distribution does not apply and, using your data, test that the median is
above 128. (3)
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j. (Extra credit) Use your data to create an approximate 90% 2-sided confidence interval for the
median.
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2.
Once again, assume that the Normal distribution applies, but assume a population standard
deviation of 16 and that we are testing whether the mean is above 128. (90% confidence level)
a. State your null and alternative hypotheses(1)
b. Find a p-value for the null hypothesis using the mean that you found in a. On the basis of your
p-value, would you reject the null hypothesis? Why? (1)
c. Create a power curve for the test. (6)
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3.
(Moore, Notz) A recent survey said that nationwide 73% of all freshman students identified being
well-off financially as an important lifetime goal. You believe that the proportion of freshman
business majors with that goal is higher than the national figure. You take a survey of a random
sample of 200 students and find that 152  a  have being well-off as an important goal, where a
is the second to last digit of your student number. If the second to last digit of your student number
is zero, a  10 . (For example, Seymour Butz’s student number is 976502, so he will add 10 and
say that x  152  10  162 )
a. Formulate your null and alternative hypotheses and do a hypothesis test with a 95% confidence
level. (2)
b. Find a p-value for the null hypothesis. (1)
c. Find the p-value for the null hypothesis if x  162  a (1)
d. (Extra credit) How would your answer to a) change if your sample of 200 came from a
population of 300? (1)
e. (Extra credit) Using a critical value of the proportion for testing your null hypothesis, create a
power curve for the test by using the alternate hypothesis and finding the power for values of
p1  .73 . (Up to 6 points)
f. Assume that p  .73 , how large a sample would you need to estimate the proportion above that
have being well-off with an error of .005? (2)
g. Use the proportion that you found in a) to create a 2-sided 95% confidence interval for the
proportion. Does it differ significantly from .73? Why? (2)
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4.
Standard deviation is often a measure of reliability. A manufacturer is providing a connector with
a mean length of 2.5 mm and is getting complaints that the connector is often too large or too
small for the intended use. The previous standard deviation to the length of the part was 0.025mm,
but the manufacturer introduces a process that should make the standard deviation smaller. A
sample of 25 items is taken which yields a sample standard deviation of 0.030  a  . To get a
take the third to last digit of your student number and multiply it by 0.001. (For example, Seymour
Butz’s student number is 976502, so he will subtract .005 and say that s  0.030  0.005  0.025 . )
a. Formulate the null and alternative hypotheses necessary to see if the goal has been achieved and
test the hypothesis using a 95% confidence level and a test ratio. (2)
b. What assumptions are necessary to perform this test? (1)
c. Try to get a rough p-value. Interpret its meaning (1.5)
d. Do a 95% two- sided confidence interval for the standard deviation (1)
e. (Extra credit) Redo 4a) using an appropriate confidence interval. (2)
f. (Extra credit) Find a critical value for s in 4a). (1)
g. The number of claims for missing baggage in a large metropolitan airport supposedly follows a
Poisson distribution with a mean of 72 per week. Assume that in a given week 92 are lost. Test
this hypothesis using a test ratio and a 95% confidence level. (2)
h. Find approximate critical values for the number of bags that could be lost in 4f. (2)
i. (Extra credit) Find the power of the test in 4g) if the average number of lost bags per week is
really 87. (3)
j. I claim that x is binomially distributed with p  .01 . Test this assertion using a 2-sided test if
there are 4 successes in 10 trials. (2)
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