Download Test 2 Review Sheet/Answer Key - Sites @ Brookdale Community

MATH 131
8/11
TEST 2 REVIEW SHEET
Links to pencast solutions to selected problems are boxed next to these problems. Written answers to all
problems can be found at the end of the review sheet.
Important terms and phrases: To locate these terms in the book, please use the index at the back
of the book.
Sample Space
Fundamental Counting Principle
Classical Probability
Permutation and Combination
Empirical Probability
Discrete/ Continuous Random Variable
Law of Large Numbers
Discrete Probability Distribution
Empirical Rule
Binomial Experiments
Chebychev’s Theorem
Normal Distribution
Conditional Probability
Standard Normal Distribution
Independent/ Dependent Events
Standard Error of the mean
Mutually Exclusive Events
Central Limit Theorem
Multiplication Rule
Normal Approximation to Binomial
Addition Rule
1. A department store sells sport shirts in three sizes (small, medium and large) and two sleeve
lengths (Full and Half).
Small
Full Sleeve 25
Half Sleeve 28
Total
53
a)
b)
c)
d)
e)
f)
g)
Medium
39
52
91
Large
56
30
86
Total
120
110
230
What is the probability that a shirt chosen from this group is a Large?
#1 SOLUTION
What is the probability that a shirt is Medium and has Full Sleeves?
What is the probability that a shirt is Small or has Half Sleeves?
What is the probability that a shirt is Large given that it has Half Sleeves?
What is the probability that a shirt is Half Sleeves given that it is a size Medium?
Are the events “Small size” and “Full Sleeves” mutually exclusive? EXPLAIN.
Are the events “Large size” and “Half Sleeves” independent? EXPLAIN. (Hint: You could use 2
of your prior answers.)
2. Of the patients examined at a local clinic, 25% had high blood pressure, 50% were overweight
and 15% had both conditions.
a) Are the two events "high blood pressure" and "overweight" mutually
#2 SOLUTION
exclusive? EXPLAIN.
b) Are the two events "high blood pressure" and "overweight" independent? EXPLAIN.
c) If one of the patients is selected at random, what is the probability that the patient has high blood
pressure or is overweight?
3. Rita is playing 3 tennis matches.
a) List the sample with the possible win and loss sequences that
she can experience for this set of three matches.
b) Is this an equally likely sample space? EXPLAIN.
1
#3 SOLUTION
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TEST 2 REVIEW SHEET
4. a) Explain clearly the difference between a combination and a permutation.
b) New Jersey has 14 four year colleges. Diana wants to apply
to 3 of them. In how many different ways can she select
#4 SOLUTION
three colleges to apply to?
5. It was a well-known rule of the sea that if a ship is sinking, then lifeboats are filled first with
women and children. Was this rule followed on Monday, April 5, 1912 when the Titanic sank?
Survived
Died
Total
a)
b)
c)
d)
e)
f)
g)
h)
i)
Men
332
1360
1692
Women
318
104
422
Boys
29
35
64
Girls
27
18
45
Total
706
1517
2223
If you select a passenger at random from this group, find the probability that the passenger was a
Woman.
Survivor.
Survivor, given that the passenger was a Woman.
Survivor, given that the passenger was a Man.
Boy or a Survivor.
Man and someone who Died.
Do you think Women had better survival rates on the Titanic than Men? (Hint: use your answer
in part c and d)
Are the events “Men” and “Survivor” independent events? Show the probabilities used to draw
your conclusion.
Are the events “Women” and “Girls” mutually exclusive events? How do you know?
6. Brookdale offers different summer camp programs for children. Next year they expect 75% of
the campers to register for sports camp, 25% to register for arts camp, and 5% to register for
both.
# 6 SOLUTION
a) Are the two events “arts camp” and “sports camp” mutually exclusive?
EXPLAIN.
b) Find the probability that a camper selected at random goes to arts camp
or sports camp.
7. A citizens action committee in NJ is comprised of six Democrats, eight Republicans, and three
Conservatives.
a) If one member is randomly selected, find the probability of getting a Conservative.
b) If one member is selected at random, find the probability of
#7 SOLUTION
getting a Republican or a Conservative.
c) If two different people are selected at random from this committee,
find the probability that they are both Democrats.
8. Ten equally qualified people apply for a job and three will be interviewed by the Manager. How
many different groups of three people can be selected for the interview?
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TEST 2 REVIEW SHEET
9. A state lottery involves the random selection of six numbers between 1 and 44. To win the
lottery, you must select the six winning numbers but not necessarily in any specific order. If you
select one set of six numbers, what is the probability it will be the winning set of numbers?
10. How many ways can seven people be seated around a table?
11. According to the Census Bureau’s Current Population Report, 1/3 of American children (under
18 years old) are not living with both parents. If 5 American children are selected at random,
find the probability that exactly two of them are not living with both parents. (Use the
appropriate formula to find the probability. Leave answer in fraction form.)
12. Seventy-five percent of households in the United States subscribe to cable TV. If you randomly
select 5 households and ask each if they subscribe to cable TV, find the probability that
a) at least four houses have cable.
#12 SOLUTION
b) at most 2 houses have cable.
c) all of the houses have cable.
13. The rates of on-time flights for commercial jets are continuously tracked by the U. S.
Department of Transportation. Recently, Southwest Air had the best rate with 80% of its flights
arriving on time. If 15 Southwest flights are randomly observed, find the probability that
a) none are on time
b) all are on time
#13 SOLUTION
c) at least 10 are on time
d) no more than 4 are on time
14. A publisher introduces a new weekly magazine for teenagers. The
company’s marketers estimate that the sales x (in thousands) will
be approximated by the distribution on the right.
a) Check whether the given distribution is a probability distribution.
b) If it is a probability distribution, find the mean for this
distribution.
c) If it is a probability distribution, find the standard deviation.
x
10
15
20
25
P(x)
.200
.300
.150
.350
#14 SOLUTION
15. A package of frozen vegetables has a label that says “contents 32 oz.” However, the company
that produces these packages knows that the weights are normally distributed with mean 32 oz.
and standard deviation 2 oz. If a package is chosen at random, what is the probability that it will
weigh?
a) more than 35 oz.?
#15 SOLUTION
b) less than 30 oz.?
c) between 30 and 35 oz.?
16. Suppose that the test scores for a college entrance exam are normally distributed with a mean
of 450 and a standard deviation of 100.
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a) What percentage of those taking the exam score between 350 and 450?
b) A student scoring above 400 is automatically admitted. What percent score above 400?
c) The upper 5% receive scholarships. What score must a student make on the exam to receive a
scholarship?
17. The heights of adult males are normally distributed with a mean height of 70 inches and a
standard deviation of 2.6 inches. How high should the doorway be
#17 SOLUTION
constructed so that 90% of the men can pass through it without
having to bend?
18. An advertising agency was hired to introduce a new product. It is claimed that after its
campaign 40% of all consumers were familiar with the product.
a) Is this a binomial experiment? EXPLAIN.
#18 SOLUTION
b) Find that the probability that out of 500 consumers, 215 or more of
them will become familiar with the product because of the campaign.
19. Thirty-three percent of adults consider public schools good at preparing students for college.
You randomly selected 40 adults and ask them if they think public schools are good at preparing
students for college. Decide whether you can use the normal distribution to approximate the
binomial distribution. Find the probability that between 15 to 17 inclusive adults will say yes.
20. Currently, most of U.S. companies test new employees for drugs, and 3.8% of those
prospective employees test positive (data based on American
#20 SOLUTION
Management Association). The Alpha Electronics company
tests 150 prospective employees. Estimate the probability of
at most 10 of them testing positive for drugs.
21. The heights of college age men are known to be normally distributed with a mean of 68 in. and a
standard deviation of 3 in.
a) What is the probability that the mean height of a random sample of nine college men is between
67 and 69 inches?
b) What is the probability that the mean height of a random sample of 100 college men is between
67 and 69 inches?
c) Compare your answers to parts a and b. Was the probability in part b much higher? Why would
you expect this?
d) Name the theorem you used to solve this problem.
e) List the criteria to apply the theorem.
22. According to the book "Standing up to the SAT", the average SAT score for Native American
males is 852. Assume that the standard deviation is 120 points.
a) Do you think that the distribution of the SAT scores is normal for
#22 SOLUTION
any group? Sketch what the distribution would look like if it was
normal. Use this in your explanation.
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b) If a random sample of 50 Native Americans is surveyed, what is the probability that their mean
SAT score is more than 900?
c) Could you have answered the question if the sample size was 25 instead of 50? EXPLAIN your
answer, mentioning any appropriate theorem.
23. The amount of coffee dispensed by a vending machine is normally distributed with a mean of
7.8 oz. and a standard deviation of 0.1 oz. The size cup used in this machine is an 8 oz. cup.
a) What percentage of cups will overflow with coffee?
b) When coffee overflows a cup, the company loses money.
#23 SOLUTION
What could the company do to further limit the amount of
overflow?
24. The monthly utility bills in a certain city are normally distributed, with a mean of $100 and a
standard deviation of $12. A utility bill is randomly selected.
a) Find the probability that the utility bill is less than $80.
#24 SOLUTION
b) Find the probability that the utility bill is between $80 and $115.
c) Find the probability that the utility bill is more than $115.
25. a) What is the probability of an event that is certain to occur?
b) What is the probability of an impossible event?
c) On a true/false problem, what is the probability of answering a question correctly if you
make a random guess?
#25 SOLUTION
d) On a multiple-choice test with five possible answers for each
question, what is the probability of answering a question correctly if
you make a random guess?
26. In a research poll, respondents were asked how a fruitcake should be used.
Use
Doorstop
Birdfeed
Landfill
Gift
Other
Number of respondents, f
50
25
35
10
5
#26 SOLUTION
a) If a respondent is picked at random, what is the probability that they would use the fruitcake
as a doorstop?
b) If a respondent is picked at random, what is the probability that they would not use it as
birdfeed?
27. Explain, clearly and in your own words, the Law of Large Numbers.
5
#27 SOLUTION
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TEST 2 REVIEW SHEET
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2
3
MATH 131
11.
5 C2 (1/3) (2/3) = 80/243
Test 2 Review Sheet/Answer Key
12. a) P(4) + P(5) = .633
1. a)
b)
c)
d)
e)
f)
g)
2. a)
b)
c)
3. a)
b)
P(0) = 0.000
P(15) = 0.0352
P(10) + P(11) +...+ P(15) = 0.939
P(0) + P(1)...+ P(4) = 0.000
14. a)
Yes it is. Each probability is between 0 and 1
sum of the probabilities equals 1.
b) 18.25
c) 5.7608
15. a)
z
35 32
1.5
2
P( x 35) P( z 1.5)
1 0.9332 0.0668
WWW, WWL, WLW, WLL
LWW, LWL, LLW, LLL
Maybe. Is she a 50/50 tennis player? Is she as
likely to win as to lose?
See Section 3.4.
5. a)
b)
c)
d)
e)
f)
g)
422/2223 = 0.1898
706/2223 = 0.3176
318/422 = 0.7536
332/1692 = 0.1962
64/2223 + 706/2223 – 29/2223 = 0.3333
1360/2223 = 0.6118
Yes. 75.4% of the women survived as
compared to only 19.6% of the men.
Not independent.
P(Man) = 1692/2223 = 0.7611 and does not
equal P(Man given Survivor) = 332/706 =
0.4702
Yes, because no one is both a Woman and a
Girl. P(Woman and Girl) = 0
i)
13. a)
b)
c)
d)
No, 15% are both.
No, P(H and O) = 0.15
0.125
= P(H)
P(O). It has been shown that
patients who are overweight are more likely
to have high blood pressure.
0.60
4. a)
b)
h)
b) P(0) + P(1) + P(2) = .104
c) P(5) = .237
86/230 = 0.3739
39/230 = 0.1696
53/230 + 110/230 – 28/230 = .5870
30/110 = 0.2727
52/91=0.5714
No, 25 shirts are both.
Comparing answers (a) and (d), you can see
P(Large) = 0.3739
0.2727 P(Large,
given Half Sleeves). Not independent.
14
C3
364
6. a)
b)
No, 5% go for both.
95%
7. a)
b)
c)
3/17=0.1765
11/17=0.6471
(6/17)(5/16) =0.1103
8.
10
9.
10.
1/7,059,052= 0.000001416
7 = 5040 = 7 P7
C3
120
6
b)
z
c)
z
30 32
2
1
P( x 30) P( z
0.1587
35 32
30 32
1.5 z
1
2
2
P(30 x 35) P( 1 z 1.5)
0.9332 0.1587 0.7745
1)
MATH 131
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TEST 2 REVIEW SHEET
c)
x
450
350 450
16. a)
z
1
100
P(350 x 450) P( 1 z 0)
0.5 0.1587 0.3413 34.13%
b)
z
1.645 100
614.5
17.
x
x
x
z
70 (2.6)(1.28)
73.3 inches
18. a) Yes, it is a binomial experiment.
See Section 4.2.
b) np 500(0.4) 200 5
nq 500(0.6) 300 5
Use the normal distribution to approximate the
binomial distribution.
np 200
400 450
0.5
100
P( x 400) 1 P( z
0.5)
1 0.3085 0.6915 69.15%
z
500(0.4)(0.6)
P( x 215) P( x 214.5)
P( z 1.32) 1 0.9066 0.0934
19.
np 40(0.33) 13.2 5
nq 40(0.67) 26.8 5
We can use the normal distribution to
approximate the binomial distribution.
np 13.2
Since
npq
7
40(0.33)(0.67)
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TEST 2 REVIEW SHEET
P(15 y 17) P(14.5 x 17.5)
P(0.44 z 1.45) 0.9265 0.67
0.2565
b)
z
z
69 68
3/ 100
67 68
3.33
3.33
3 / 100
20. Let x = prospective employees test positive.
n = 150, p(x) = 3.8% = 0.038
np 150(0.038) 5.7 5
nq 150(0.962) 144.3 5
P(67
x
69)
P( 3.33 z
3.33)
0.9996 0.0004 0.9992
c) Yes, explain.
d) Central Limit Theorem
e) See Section 5.4.
Use normal distribution to approximate the
binomial distribution.
np 5.7
22. a)
b)
npq
(150)(0.038)(0.962)
P( x 10) P( x 10.5)
P( z 2.05) 0.9798
10.5 5.7
z
2.05
150 0.038 0.962
21. a)
Use CLT because is finding the mean height
P(67 x 69) P( 1 z 1)
68
x
0.8413 0.1587 0.6826
z
69 68
3/ 9
1
z
67 68
1
3/ 9
8
Explain
n = 50 >30 use CLT
852
x
120
x
n
50
x
900 852
z
/ n 120 / 50
2.83
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P( x
TEST 2 REVIEW SHEET
900)
P( z
2.83)
1 0.9977 0.0023
c) With a sample size of 25 the CLT does not apply.
Methods of nonparametric statistics
would be needed.
23. a)
z
P( x 8)
8 7.8
0.1
P( z
2) 1 0.9772 0.0228
2.28%
c)
b) Lower the standard deviation. Set the mean lower.
Use a bigger cup.
24a)
80 100
1.67
12
P( x 80) P( z
1.67) 0.0475
z
P( x 115) P ( x 1.25)
1 P( z 1.25) 1 0.8944
0.1056
25. a) 100%
b) 0%
1
2
1
d) 20% or .2 or
5
c)
b) P(80
x 115) ?
115 100
z
1.25
12
P(80 x 115) P( 1.67
0.8944 0.0475
0.8469
50% or .5 or
2
50
=
or 40% or .4
125 5
4
25
b) 1
=
or .8 or 80%
5
125
26. a)
x 1.25)
27. See Section 3.1.
9