Review Materials for Midterm 2
Coverage: Chapters 4.1, 4.2, 4.3, 4.4, 5.1, 5.2, 6.1, 6.2, 6.3
Suggested
Extra Problems from your textbook:
Chapter 4.3: 4.46 (a-c), 4.49
Chapter 4.4: 4.55, 4.57, 4.61, 4.69
Chapter 4 Exercises (p. 365): 4.98, 4.99, 4.104(a)
Chapter 5.1: 5.7 (answer in the back is wrong, ask what is the appropriate parameter to get the right answer), 5.10, 5.14, 5.20
Chapter 5.2: 5.26, 5.29, 5.31
Chapter 6.1: 6.16, 6.20, 6.23
Chapter 6.2: 6.31, 6.32(a-c), 6.35
Chapter 6.3: 6.55
Chapter 6 Exercises (p. 496): 6.76, 6.80
Topics/Vocabulary
Sample Space
Event
Probability Rules (page 298)
Multiplication Rule
Independence
Random Variables: Discrete
& Continuous
Probability Distribution
Mean or Expected Value of
a Random Variable
Rules for means (p. 334)
Variance of a Random Variable
Rules for variances (p. 337)
Sampling distribution
Counts & proportions
Binomial mean and standard
deviation
Sample proportions, mean and
standard deviation of a sample proportion
Normal approximation (p. 383)
Mean and Standard Deviation
of a Sample mean (p. 399)
The Central Limit Theorem (p.
402)
Confidence Intervals (p.
434-445)
Significance Test (p.
453-468; 475-481)
2, not 1, Old Midterms Follow, so don’t get upset,
the real midterm will not be this long. One of your classmates requested more practice problems, so
here they are. Answers will be
posted to the web.
Midterm 2A
1. A nationally televised
news program offered people the opportunity to express their reactions to a
change in the tax laws. People
could call one number to indicate support for the change, people could call
another number to express their opposition to the change. Each call costs $1.00. During the evening, 379,275 calls
indicated support and 120,725 expressed opposition. Please construct a 95% confidence interval for the
population proportion, p, (those who support the change in the tax laws). If constructing a 95% confidence
interval is not the appropriate thing to do, please write "not
appropriate" below and then please explain why.
2.The Los Angeles Times
conducted a survey of Asian-owned businesses by randomly sampling 144
businesses from state tax records.
Assume the sample is of good quality. From their sample, the technical staff gave a 90% confidence
interval for the average age of Asian business owners of:
51.1 years ± 2.74 years
Suppose the population
standard deviation, sigma, s, is known and it is 20 years and suppose the editor
is very unhappy with the confidence interval and he told them he would like the
90% confidence interval to be no larger than ± 1.00 years of the average age. Please indicate if this is (a) possible
to do and if it is possible (b) describe how to accomplish this -- calculations
may be necessary. If it is not
possible, discuss why this is not possible
(a) Possible or not possible?
If possible, just write "possible" in the space below and continue on
to (b). If not possible, write "not possible" in the space
below, justify your response, and leave part (b) blank.
(b) It is possible and shown
below is how a 90% confidence that is no larger than ± 1.00 years of the average age could be constructed:
3. A marketing company wishes to determine the extent to
which people who currently own personal computers would be willing to
"upgrade" to handheld "communication devices" such as the
Palm Pilot (manufactured by 3Com).
Because of logistical considerations, the survey focused on households
that purchased new personal computers within the last year. A list of all such households was
obtained from manufacturers through warranty registration records, and from
this list the marketing company determined that amount paid for a computer was
normally distributed with an average of $3,100 and a standard deviation of
$1355. From this list, the company
selected a simple random sample (SRS) of size 64 and conducted in-person
household interviews. Analysis of
the sample revealed that the 64 households paid an average of $2,850 for their
personal computer (with a standard deviation of $1200) and 21% of the
households surveyed said they will purchase a handheld "communication
device" sometime in the next five years.
a. Construct a 99% confidence interval for
the population proportion, p, (the proportion who will purchase a handheld
"communication device" of the households that purchased a computer
last year).
- What is the chance of getting a sample average of
$2,850 or lower for a sample of size 64 from the population of households
that purchased a computer last year?
(continued from the previous
problem)
c. Suppose the CEO of 3Com
claims to know (he's an extra smart guy, right) that everyone would be willing
to upgrade to a Palm Pilot if the price was right. He thinks the "right
price" is an average of $259 for the product but he didn't give
information about the standard deviation.
The marketing company found that in its sample of 64, the "right
price" had an average of $235 with a standard deviation of $110. The marketing company thinks the CEO's
claim is too high and they hire you to test his claim. Please test the CEO's claim using
information from the sample.
First, state the null and alternative hypotheses (5 points). Second, if it is possible, construct a
test statistic, state the resulting p-value and render a decision using a 5%
level of significance -- do you reject his claim or not? If it is not possible
to test his claim, please indicate that it is "not possible" and
explain why.
4.
Congratulations, you're
a traveling salesperson for a large manufacturer. You make 3 calls per year on each client. Your chance of a sale each time is
75%. Let X denote the total number
of sales in a year.
a. Fill out this table
|
F(x) |
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P(x) |
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b. Suppose you have 100 clients, what is your expected
number of sales in a year?
c. What is the standard deviation for your expected
number of sales in a year?
d. Suppose you get $5,000 per sale. What are your expected earnings in a
year?
Midterm 2B
1. Congratulations,
you finished college and became a portfolio manager for a large financial
services firm. The bad news is
that your portfolio return of 3.6% (with a standard deviation of 15.5%) is well
below the historical Wall Street average return of 10.0% (with a standard
deviation of 25.8%). You are
worried since your supervisor is only willing to employ you if your return is
greater than Wall Street's . You
call "the Oracle" for help. The Oracle says two things (a) increase the number of
stocks in your portfolio from 49 to 144 and (b) use his stock selection method
because you will get an average return of 12.6% (with a standard deviation of
14.1%). Treat Oracle's selection method as if it is random.
a. (5 points) What are the appropriate null and
alternative hypotheses to test whether
implementing the Oracle's
recommendations will help your portfolio return more than the historical Wall
Street average?
b. (10 points) What is the appropriate test for part
(a)? If it is possible to conduct
a test, do so in the space below and give a resulting p-value. Show your work and clearly identify the
resulting p-value.
c. (5 points) If you performed a test in part (b) above,
what are your conclusions? Is
there enough evidence for you to
switch to Oracle's stock selection method? Use a 5% level of significance to help you render your
decision. Explain. Be brief.
2. The CEO of company A bids
on consulting jobs so that if awarded the job, company A will make a $200,000
profit on that job. The CEO of
company B bids on consulting jobs so that if awarded the job, company B will
make a $600,000 profit. Each
company has a probability distribution of the number of jobs the company is
awarded per year…unfortunately, there is a little bit of missing
information:
a. Fill in the missing
information (2 points each company, 4 points total)
Company A
|
F(x) |
0 |
1 |
2 |
3 |
4 |
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P(x) |
.01 |
|
.53 |
.32 |
.00 |
Company B
|
F(x) |
0 |
1 |
2 |
3 |
4 |
|
P(x) |
.00 |
.37 |
|
.22 |
.04 |
b.
What is the expected
profit for each company? (3 points each company, 6 points total)
c.
Find the standard
deviation of the number of consulting jobs awarded per year for each company.
(5 points each company)
d.
What is the difference
between the number of consulting jobs awarded per year for the two companies? What is the standard deviation of this
difference? Assume the number of
jobs awarded to the companies are independent. (10 points)
3. A food-products
company conducted a market study by randomly sampling and interviewing 400
consumers to determine which brand of snacks (e.g. chips, pretzels) they
prefer. Suppose 117 consumers or
.293 were found to prefer the company's brand.
(a)
(5 points) Construct a
90% confidence interval for the proportion of consumers who prefer the
company's brand.
(b)
The Director of Marketing
walks by and is unhappy with your results. She
tells you that she wants a
better estimate. Specifically she
wants you to be within .010 of the proportion of consumers who prefer the
company's brand, 95 out of 100 times. Is there anything you can do to honor her
request? Answer yes or no and if
you answer yes, please tell us what you can do and supply a numerical
solution. If you answer no,
explain why you cannot do this for her. (5 points)
4. (3 points each) Indicate whether the statement is true
or false about the following confidence interval which was generated from a
sample of 441 with a sample mean of 1.1 and sample standard deviation of 11.7:
1.1
± 2.576(11.7/Ö441) = 1.1 ± 1.44
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4. |
True |
False |
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A |
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Increasing the confidence
level will decrease the width of the interval. |
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B |
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In 99% of all samples the
true mean m
will lie within |
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C |
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There is a 99% chance that
the population mean lies in the interval (-0.34,2.54) |
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D |
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This is a 99% confidence
interval for the population mean |
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E |
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Decreasing the sample size
will decrease the width of the interval |
