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ECO252 QBA2
FIRST HOUR EXAM
February 28 2005
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 1.5, 6
1. P0  x  21 .00 
2. P 7.00  x  1.50 
3. Px  10 .22 
4.
x.08
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II. (5 points-2 point penalty for not trying part a.)
(Mansfield) A random sample is taken of the length in feet of aluminum foil rolls. The following
data is found. (Recomputing what I’ve done for you is a great way to waste time.)
x
x2
1
2
3
4
5
6
7
Sum
74.88
75.86
74.81
74.28
74.35
73.41
74.66
522.25
5607.0144
5754.7396
5596.5361
5517.5184
5527.9225
5389.0281
5574.1156
38966.8747
a. Compute the sample standard deviation, s , of the waiting times. Show your work! (2)
b. Compute a 99% confidence interval for the population mean,  . (2)
c. Is the population mean significantly different from 75.8 ft? (1)
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III. Do all of the following Problems (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.
When a p-value is smaller than a significance level
a) A type one error has been committed
b) A type two error has been committed
c) The null hypothesis is rejected
d) The alterative hypothesis is rejected
e) The critical value is correct.
2.
The t distribution should be used when the parent (underlying) population
a) Is Normal, the population standard deviation is unknown and we are testing a mean.
b) Is Normal, the population standard deviation is known and we are testing a mean.
c) Is Normal, the mean of the population is unknown and we are testing a mean.
d) Is binomial and we are testing for a proportion.
e) The t distribution should be used in all of these cases.
3.
The Normal distribution can be used in all of the cases below except when:
a) We are testing a mean, the population standard deviation is unknown and the sample is
large.
b) We are testing a proportion and the sample is large.
c) We are testing a variance and the sample is large.
d) We are testing the mean of a Poisson distribution and the sample is large.
e) All of the above are cases when the Normal distribution can be used.
[6]
4.
(Lange) The state wants to estimate the proportion of the labor force that was unemployed in
North Hotzeplotz and wants to be 99% confident that their estimate is within 5% (written 0.05) of
the population proportion. If the proportion is probably about 15%, how large a sample is needed?
(3)
[9]
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5.
I wish to do a test to see if the average level of satisfaction of my employees is above 75 on a zero
to 100 scale. I take a survey of 30 of my 100 employees and get a mean of 76. I assume that the
data is Normally distributed with a standard deviation of 8. What are my null and alternate
hypotheses?
H 0 :   75 and H 1 :   75
b) H 0 :   75 and H1 :   75
c) H 0 :   75 and H 1 :   75
d) H 0 :   75 and H 1 :   75
e) H 0 :   75 and H 1 :   75
f) H 0 :   75 and H 1 :   75
g) None of the above.
a)
6.
I wish to do a test to see if the average level of satisfaction of my employees is above 75 on a zero
to 100 scale. I take a survey of 30 of my 100 employees and get a mean of 76. I assume that the
data is Normally distributed with a standard deviation of 8. Assume that your null and alternate
hypotheses in question 5 are correct and that your confidence level is 92%. Find a critical value for
the sample mean. Show clearly what formulas you are using. (3)
[14]
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7.
(Dummeldinger)A movie rental chain is considering opening a new outlet. The corporation will
open an outlet only if more than 5000 out of the 20000 households in the area have DVD players.
It randomly selects 300 households and finds that 96 have DVD players. What are our null and
alternative hypotheses?
a) H 0 :   5000 and H1 :   5000
b) H 0 : p  5000 and H1 : p  5000
c)
d)
e)
f)
8.
: p  .32 and H1 : p  .32
: p  .25 and H1 : p  .25
:   .25 and H1 :   .25
:   5000 and H1 :   5000
[16]
We wish to determine if the median income in an area exceeds $40000. A random sample of 250
households was selected. 104 had incomes above $40000. Let p be the proportion with incomes
above $40000. Our null hypotheses include (3):
a) H 0 :   40000
b)
c)
d)
e)
f)
g)
h)
i)
9.
H0
H0
H0
H0
H 0 :   40000
H 0 : p  0.5
H 0 : p  0.5
H 0 : p  0.5
H 0 :   40000
H 0 :   40000
H 0 :   0.5
H 0 :   0.5
[19]
We wish to determine if the median income in an area exceeds $40000. A random sample of 250
households was selected. 104 had incomes above $40000. Let p be the proportion with incomes
above $40000. Assume that your null hypotheses in 8) are correct and test the hypothesis.
a) Using a test ratio and a p-value. (2).
b) Using a critical value for x , p, s or the median as appropriate. (2).
c) Using a confidence interval. (2, 3 or 5 depending on your level of chutzpah).
d) (Extra Credit) Redo a), b) or c) assuming that the sample of 250 came from a population
of 2000.
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Blank page for calculations. If you use any other paper for calculations, turn it in – it can only help you.
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252y0511 2/17/05 (Open in ‘Print Layout’ format)
ECO252 QBA2
FIRST EXAM
February 28 2005
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 are not usually acceptable.
1.
(Dummeldinger) You are an automobile manufacturer and the EPA has just estimated that your
2005 Prejector model gets 35 miles per gallon on the highway. You wish to prove that the
Prejector gets more than 35 mpg. 50 of the current model are tested with the results below. To
personalize the data below take the last digit of your student number, divide it by 10 and add it to
the numbers below. (For example, Seymour Butz’s student number is 976502, so he will add 0.20
and change the data to 44.64, 48.04, 37.57 etc. – but see the hint below, you do not need to write
down the numbers that you are using, just your computations.)
Miles per gallon
44.44
47.84
34.59
32.02
35.61
42.56
40.92
33.56
44.26
21.41
42.70
44.70
37.37
36.27
35.80
46.52
24.56
38.24
2
39.57
23.55
51.16
45.40
37.59
41.49
21.05
44.28
29.14
29.54
34.07
48.02
43.41
41.86
42.31
23.98
24.03
35.20
27.58
41.13
32.18
39.03
44.44
36.78
35.47
33.88
 x  1860 .17 ,  x  71904 .65,  x  a    x   na,
  x   2a x  na
Hint: n  50 ,
 x  a 
41.37
33.47
43.45
35.72
43.58
33.07
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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)
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, is the EPA right? Why? (1)
h. Assume that the Normal distribution does not apply and, using the data as given above, test that
the median is above 35. (3)
i. (Extra credit) Again, use the data as given and do 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 7 and that we are testing whether the mean is below 36 mpg. (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. (1)
c. Create a power curve for the test. (6)
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3.
a. Assume that you are testing the hypothesis   36 using the original data. Let p be the
proportion of the data above 36, so that, according to the outline, your alternate hypothesis is
p  .5. Using a 99% confidence level find a critical value for p , how many items in the sample of
50 would have to be above 36 for you to reject the null hypothesis? (This answer should either say
‘between 0 and ?’ or ‘between ? and 50.’) (2)
b. Using the proportion of numbers above 36 in the original data, find a p-value for the null
hypothesis. (1)
c. (Extra credit) Create a power curve for the test by using the alternate hypothesis in b and finding
the power for other values of p1 . (up to 6)
d. Assume that p  .5 , how large a sample would you need to estimate the proportion above 36
with an error of .01? How much would you cut down the sample size if you used the proportion
that you actually found? Illustrate how much the required sample size would fall if you lowered
the confidence level. (3)
e. Use the proportion that you found in 3b) to create a 2-sided confidence interval for the
proportion above 36. Does it differ significantly from .5? Why? (2)
4.
a. Take the standard deviation that you found in 1), add the same quantity that you added in part 1)
to it. (For example, Seymour Butz’s student number is 976502 and he found s  7.12 , so he will
add 0.20 to it and use 7.32.)
b. Test the hypothesis that the standard deviation is 6. (99% confidence level) Use a test ratio. (2)
Find a p-value for your answer in 4a). (1)
c. Do a 99% confidence interval for the standard deviation (2)
d. (Extra credit) Redo 4a) using an appropriate confidence interval. (2)
e. (Extra credit) Find critical values for s in 4a). (1)
f. A bank's average default rate on loans is supposedly 7 per month. In the first month there are
13 defaults. Test the first assertion assuming a Poisson distribution. Use a two-sided test with a
1% significance level. (2)
g. In 4f) find what values of x (the number of defaults in the first month) would enable you not to
reject the null hypothesis. (2)
h. (Extra credit) Assume that the bank, in fact, has an average default rate on loans of 9 per
month, what is the probability that you will fail to reject your null hypothesis that the mean is 7,
using the ‘accept’ zone that you found in g)?
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