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ECO252 QBA2
FIRST HOUR EXAM
February 23, 2007
Version 1
Name ________________
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 2, 7
1. P 23 .4  x  2
2. P 7  x  7 
3. Px  0
4. x.085 (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.) Edwin Mansfield tells us that in order to compute
information about the average expenditure by honors students at Semiconscious State University, a sample
of four students’ expenditures is taken. The results are as below. I should not have to tell you that parts b, c,
and f require hypothesis tests.
x
192
150
168
90
a. Compute the sample standard deviation, s , of expenditures. Show your work! (2)
b. Is the population mean significantly above 130? (Use a 90% confidence level)? Show your
work! (3)
c. Redo b) when you find out that there are only 10 honors students at Semiconscious State
University. (2)
d. (Extra Credit) Find an approximate p value for your null hypothesis. (2)
e. Assume that the population standard deviation is 40 and create an 83% two-sided confidence
interval for the mean. (2)
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f. (Extra Credit) Given the data, test the hypothesis that the population standard deviation is above
40?
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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.
1.Which of the following is a Type 1 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. In a survey, 540 out of 1000 respondents indicated that they would rather have $150 than a day off.
You believe that a majority (over 50%) will prefer the money to the day off. Do the data support your
opinion? Your hypotheses are:
a) H 0 : p  .50 , H 0 : p  .50
b) H 0 : p  .50 , H 0 : p  .50
c) H 0 : p  .50 , H 0 : p  .50
d) H 0 : p  .50 , H 0 : p  .50
e) H 0 : p  .50 , H 0 : p  .50
f) None of the above.
3. The chi-squared distribution is used in which of the following circumstances.
a) We are testing a single mean, the underlying distribution is Normal and the population
variance is unknown.
b) We are testing a single variance, the underlying distribution is Normal and the population
variance is unknown.
c) We are testing a single proportion, the underlying distribution Binomial and we have a
large sample.
d) We are testing a single median, the underlying distribution is Normal and the population
variance is unknown.
e) None of the above.
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4. According to Dummeldinger, The Florida Association of Retired Teachers (FART) is thinking of
changing its name. It takes a survey of 60 members and finds that many wish to change the name.
Assuming that the alternative hypothesis is H1 : p  .6 and that   .01 , how many people (what
proportion of 60) would be required to reject the null hypothesis? (Show your work.) (3) [9]
5. According to Dummeldinger, The Florida Association of Retired Teachers (FART) is thinking of
changing its name. It takes a survey of 60 members and finds that 45 wish to change the name. Assuming
that the alternative hypothesis is p  .6 , what is the p-value for the null hypothesis? (3) [12]
6. Given the numbers in problem 5) create a 95% two sided confidence interval for the proportion that
favors a change. (3)
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7. Assuming that about 60% of the old FART members favor a name change, how many members would
we have to poll to know what the proportion is within ..01 ? (2)
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8. While we are playing with old folks in Florida, assume that a Florida town has had a median age of 55.
Because they are afraid that the median age is now higher the planning board takes a sample of 200 people.
How many of the 200 people would have to be over 55 for the planning board to conclude that the median
has risen? (3)
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Blank page for calculations.
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ECO252 QBA2
FIRST EXAM
February 23, 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. 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). All problems are based on Anderson, Sweeny and Williams except 4f and 4g.
1.
The average sales of a grocery store have been $80000 per day. After new advertising policies are
adopted, sales in thousands of dollars are tabulated for 64 days. Test to see if the average sales are
now above 80000.
x
69.07
109.29
84.73
75.51
78.99
72.72
76.23
52.80
81.67
88.30
86.43
81.79
68.74
91.19
83.13
87.95
72.69
75.33
103.18
64.20
92.14
87.70
78.09
74.41
72.73
86.89
90.65
86.96
68.47
85.43
85.03
92.67
82.26
83.34
63.91
83.54
72.60
78.85
66.50
69.59
83.59
80.67
65.65
91.22
81.44
82.75
91.43
91.48
87.44
83.41
93.22
91.05
82.13
91.84
97.30
75.19
94.47
96.50
71.65
91.50
76.37
95.71
91.49
86.80
To personalize the data below take the last digit of your student number, divide it by 10 and add it to the
numbers above. 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 69.27, 81.87, 72.89 etc. – but see the
hint below, you do not need to write down all the numbers that you are using, just your computations.)
Hint - if you use the computational formula: For the original numbers n  64 ,
and
x
2
 x  5280 .00
 442387 .8608 . 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 above 82 (thousand) and your sample standard deviation
should be above 10 (thousand))
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, have sales increased? Why? (1)
h. How do your conclusions change if the random sample of 64 days is taken from a population of
100 days? (2)
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i. Assume that the Normal distribution does not apply and, using your data, test that the median is
above 80000. (3)
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j. (Extra credit) Use your data to get an approximate 99% 2-sided confidence interval for the
median.
2.
Once again, assume that the Normal distribution applies, but assume a population standard
deviation of 10 (thousand) and that we are testing whether the mean is above 80 (thousand). (99%
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.
Nationwide 16.5% of all CEOs have advanced degrees. You take a survey of a random sample of
160 CEOs in your city and find that 24  a  have advanced degrees, where a is the second to last
digit of your student number. (For example, Seymour Butz’s student number is 976512, so he will
subtract one and say that x  24  1  23 .) Label your solution ‘Version a ,’ where a is the
number that you are using. Is the fraction of executives that have advanced degrees in your city
significantly less than the national percentage?
a. Formulate your null and alternative hypotheses and do a hypothesis test with a 99% confidence
level. (2)
b. Find a p-value for the null hypothesis. (1)
c. (Extra credit) How would your answer to a) change if your sample of 160 came from a
population of 200? (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  .165 . (Up to 6 points)
d. Assume that the proportion of CEOs in your city is the observed proportion in part a, how large
a sample would you need to estimate the proportion above that have Advanced degrees with an
error of .001? (2)
e. Use the proportion that you found in a) to create a 2-sided 99% confidence interval for the
proportion. Does it differ significantly from .165? Why? (2)
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4.
Standard deviation is often a measure of reliability. A manufacturer is claiming that the equipment
in use to fill bottles fills the bottles so that the standard deviation is 0.1 oz or less. You take a
sample of 20 bottles and get a sample standard deviation of 0.110  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 add .005 and say that s  0.110  0.005  0.115 . ) Label your
solution Version a .
a. Formulate the null and alternative hypotheses necessary to see if the standard deviation is not
more than 0.1 and test the hypothesis using a 99% 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)
d. Do a 99% 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. Do a. again assuming that you took a sample of 110 bottles. (2)
f. A machine supposedly requires adjustment at most once a month. Last month, however it
needed to be adjusted 3 times. Assuming that the Poisson distribution applies and using a 5%
significance level, is there something wrong? (2)
g. Over a period of 5 years the average number of accidents per year on a stretch of highway was
30. The speed limit was reduced to 45 mph and there were only 4 accidents in a 3-month period.
Use the Poisson distribution to test if there has been a significant reduction in accidents. Use a 5%
significance level. (2)
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