252y061t 10/10/06 (Open in ‘Print Layout’ format)
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 139.80, 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 get an approximate 90% 2-sided confidence interval for the
median.
252y061t 10/10/06 (Open in ‘Print Layout’ format)
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 .) Label your solution ‘Version a ,’ where a is the number that you
are using.
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)
c. (Extra credit) How would your answer to a) change if your sample of 200 came from a
population of 300? (1)
d. (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)
d. 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)
e. 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)
c. Do a 95% two- sided confidence interval for the standard deviation (1)
d. (Extra credit) Redo 4a) using an appropriate confidence interval. (2)
e. (Extra credit) Find a critical value for s in 4a). (1)
f. 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)
g. Find approximate critical values for the number of bags that could be lost in 4f. (2)
h. (Extra credit) Find the power of the test in 4g) if the average number of lost bags per week is
really 87. (3)
i. 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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252y061t 10/10/06 (Open in ‘Print Layout’ format)