Download Statistics Final Exam Review

Statistics Final Exam Review
Chapters 3, 4, 5
Practice Problems with Solutions
Chapter 5 Practice Problems
1) Find the area under the standard normal curve between z = 0 and z = 3. 1
A) 0.9987
B) 0.4987
Normalcdf(0,3.1) = 0.4990
C) 0.0010
D) 0.4641
B
2) Find the area under the standard normal curve to the left of z = 1.25.
A) 0.1056
B) 0.8944
C) 0.2318
Normalcdf(-10000,1.25) = 0.8944
D) 0.7682
B
Chapter 5 Practice Problems
3) Find the area of the indicated region under the standard normal curve.
A) 0.0968
B) 0.0823
C) 0.9032
D) 0.9177
Normalcdf(-1.30,1.0000) = 0.9032
C
4) For the standard normal curve, find the z-score that corresponds to the
first quartile.
A) 0.67
B) -0.23
C) 0.77
D) -0.67
The first quartile means that P = 0.2500. The z-score that corresponds to
this percentage from the Normal Distribution chart is – 0.67. The answer
is D.
Chapter 5 Practice Problems
5) IQ test scores are normally distributed with a mean of 99 and a
standard deviation of 11. An individual's IQ score is found to be 109.
Find the z-score corresponding to this value.
A) 0.91
B) 1.10
C) -1.10
D) -0.91
Given that the value of  = 99,  = 11, and x = 109, we can solve for
the value of z using the equation z = (x - )/  . Z = 0.91 A
6) An airline knows from experience that the distribution of the number of
suitcases that get lost each week on a certain route is approximately
normal with μ = 15.5 and  = 3.6. What is the probability that during a
given week the airline will lose between 10 and 20 suitcases?
A) 0.3944
B) 0.8314
C) 0.1056
Normalcdf(10,20,15.5,3.6) = 0.8311
B
D) 0.4040
Chapter 5 Practice Problems
7.
With 5% of the data not included on either side, we look up the percent
value of .0500 on the chart and we get that z = -1.645. The z value for the
other side of the chart will be +1.645. The solution is B.
8.
Using the equation x = z + , we see that x = (15)(2.33) + 100 = 134.95.
The correct choice will be D.
Chapter 5 Practice Problems
9.
Solve for the value of z for each of the situations and then compare the results;
the larger of the two z scores will have performed better. The first z-value will
be z = (75 – 65)/8 which equals 1.25. The second z-value will be z = (75 – 70)/4 which
equals 1.25. Since both z scores are the same the correct choice will be C.
Chapter 4 Practice Problems
10. State whether the variable is discrete or continuous. The number of cups of
coffee sold in a cafeteria during lunch
A) discrete
B) continuous
A) discrete
11. State whether the variable is discrete or continuous. The blood pressures of a
group of students the day before their final exam
A) continuous
B) discrete
A) continuous
Chapter 4 Practice Problems
12) The random variable x represents the number of cars per household in a town
of 1000 households. Find the probability of randomly selecting a household that
has less than two cars.
A) 0.809
B) 0.125
C) 0.553
D) 0.428
Cars Households
0
125
1
428
2
256
3
108
4
83
P(x) = (428 + 125)/1000 = 0.553
C) 0.553
Chapter 4 Practice Problems
13) The random variable x represents the number of credit cards that adults have
along with the corresponding probabilities. Find the mean and standard deviation.
A) mean: 1.30; standard deviation: 0.44 B) mean: 1.23; standard deviation: 0.66
C) mean: 1.23; standard deviation: 0.44 D) mean: 1.30; standard deviation: 0.32
x
0
1
2
3
4
P(x)
0.07
0.68
0.21
0.03
0.01

x
P(x)
xP(x)
(x - )
(x - )2
(x - )2P(x)
0
0.07
0.00
-1.23
1.513
0.106
1
0.68
0.68
-0.23
.053
0.036
2
0.21
0.42
0.77
.593
0.125
3
0.03
0.09
1.77
3.133
0.094
4
0.01
0.04
2.77
7.673
0.077
Total
1.00
1.23
0.4.38
2
(
x


)
P( x)  0.438  0.66

B) mean: 1.23; standard deviation: 0.66
Chapter 4 Practice Problems
14) At a raffle, 10,000 tickets are sold at $5 each for three prizes valued at
$4,800, $1,200, and $400. What is the expected value of one ticket?
A) $4.36
B) -$4.36
C) $0.64
D) -$0.64
B) -$4.36
1
1
1
xP
(
x
)

(

$4,800)

(

$1,
200)

(
 $400) 

10, 000
9,999
9,998
999
(
 $5)  $0.48  $0.12  $0.04  ($5.00)  $4.36
10000
15) A sports analyst records the winners of NASCAR Winston Cup races for a
recent season. The random variable x represents the races won by a driver in
one season. Use the frequency distribution to construct a probability distribution.
Wins
1234567
Drivers 12 2 0 2 0 0 1
x
1
P(x) 0.71
2
0.12
3
0
4
0.12
5
0
6
0
7
0.06
Chapter 4 Practice Problems
16) Determine whether the distribution represents a probability distribution. If not,
identify any requirements that are not satisfied.
x
1
2
3
4
5
P(x)
0.2
0.2
0.2
0.2
0.2
It is a probability distribution, because each value is between
0 and 1 and the sum of P(x) = 1.
17. According to government data, the probability that a woman between the ages
of 25 and 29 was never married is 40%. In a random survey of 10 women in this
age group, what is the probability that at least eight were married?
P(8) + P(9) +P(10) = 0.1209 + .0403 + .0060 = .0.1672
B) 0.167
Binompdf(10,.60,8) = 0.1209; Binompdf(10,.60,9) = 0.0403; Binompdf(10,.60,10) =
0.0060
Chapter 4 Practice Problems
17) The probability that a tennis set will go to a tiebreaker is 18%. In 60 randomly
selected tennis sets, what is the mean and the standard deviation of the number of
tiebreakers?
A) mean: 10.8; standard deviation: 2.98 B) mean: 10.2; standard deviation: 2.98
B) C) mean: 10.8; standard deviation: 3.29 D) mean: 10.2; standard deviation: 3.29
Mean: μ = np = 60 x 0.18 = 10.8
Standard Deviation: =npq
= 60x.18x.82 = 2.98
A) mean: 10.8; standard deviation: 2.98
Chapter 4 Practice Problems
18) Fifty-seven percent of families say that their children have an influence on their
vacation plans. Consider a sample of eight families who are asked if their children
influence their vacation plans. Identify the values of n, p, and q, and list the
possible values of the random variable x.
n = 8; p = 0.57; q = 0.43; x = 0, 1, 2, 3, 4, 5, 6, 7, 8
19) You observe the gender of the next 100 babies born at a local hospital. You
count the number of girls born. Identify the values of n, p, and q, and list the
possible values of the random variable x
n = 100; p = 0.5; q = 0.5; x = 0, 1, 2, . . ., 99, 100
Chapter 4 Practice Problems
20. A statistics professor finds that when he schedules an office hour at the 10:30
a.m. time slot, an average of three students arrive. Use the Poisson distribution to
find the probability that in a randomly selected office hour in the 10:30 a.m. time
slot exactly five students will arrive.
A) 0.0137
3) D
B) 0.0519
 = 3; x = 5;
P( x) 
C) 0.0070
 x e 
D) 0.1008
35 e3

 0.1008
x!
5!
Poisson(3,5) = 0.1008
Chapter 4 Practice Problems
21. A mail-order company receives an average of five orders per 500 solicitations.
If it sends out 100 advertisements, find the probability of receiving at least two
orders. Use the Poisson distribution.
A) 0.1839
5) D
B) 0.9596
C) 0.9048
D) 0.2642
 x e 
10 e 1
P(0) 

 0.3679
x!
0!
 x e   11 e 1
P(1) 

 0.3679
x!
1!
P( x  2)  1  (0.3679  .03679)  0.2642
Poisson(1,0) = 0.3679; Poisson(1,1) = 0.3679; P(0) +P(1)= 2(0.3679)= 0.7358
P( x  2) = 1 – 0.7358 = 0.2642
Chapter 4 Practice Problems
22. Given: The probability that a federal income tax return is filled out incorrectly
with an error in favor of the taxpayer is 20%. Question: What is the probability type
that of the ten tax returns randomly selected for an audit, three returns will contain
only errors favoring the taxpayer?
A) Poisson
7) C
B) geometric
C) binomial
1.
The experiment is repeated for a fixed number of trials, where
each trial is independent of other trials.
2.
There are only two possible outcomes of interest for each trial.
The outcomes can be classified as a success (S) or as a
failure (F).
3.
The probability of a success P(S) is the same for each trial.
4.
The random variable x counts the number of successful trials.
Chapter 4 Practice Problems
23 Given: The probability that a federal income tax return is filled out incorrectly
with an error in favor of the taxpayer is 20%. Question: What is the probability
type that when the ten tax returns are randomly selected for an audit, the sixth
return will contain only errors favoring the taxpayer?
A) Poisson
B) binomial
C) geometric
8) C
Geometric distribution
A discrete probability distribution. Satisfies the following conditions
a) A trial is repeated until a success occurs.
b) The repeated trials are independent of each other.
c) The probability of success p is constant for each trial.
Chapter 4 Practice Problems

Binomial



Geometric



repeated for a fix # of trials
Random variable x, counts the # of successes out of n trials
repeated until a successful outcome occurs
Random variable x, is the # of where the first success occurs
Poisson


Counting the # times an event occurs for a specified interval
The number of occurrences of an event in a specified interval
is independent of the # of occurrences of the event in other
specified intervals
Chapter 4 Practice Problems
  np
Mean

Binomial
 2  npq
Variance
  npq
Standard Deviation


Geometric
Mean
Variance
Standard Deviation

Poisson
1
p
2 
q
p2

q
p2
Mean
  mean # occurrences
Variance
2  
Standard Deviation
 
Chapter 4 Practice Problems
24. Assume the probability that you will make a sale on any given telephone call
is 0.19. Find the probability that you (a) make your first sale on the fifth call,
(b) make your first sale on the first, second, or third call, and
(c) do not make a sale on the first three calls.
x = # of trials for 1st success
p = probability of success
q = probability of failure
(a) geometpdf(.19,5) = 0.082
(b) geometpdf(.19,1) + geometpdf(.19,2) +
geometpdf(.19,3) = 0.19 + 0.154 + 0.125 = 0.469
P(x) = p(q) x-1
(c) No sales on the first 3 calls = Q(1) + Q(2) + Q(3) = 1 – 0.469 = 0.531
Chapter 4 Practice Problems
24. During a 36-year period, lightning killed 2457 people in the United States.
Assume that this rate holds true today and is constant throughout the year.
Find the probability that tomorrow (a) no one in the United States will be
struck and killed by lightning, (b) one person will be struck and killed,
(c) more than one person will be struck and killed.
x = # occurrences in a
 = 2457/13140 = 0.1870
given time or space
= average # of
occurrences in a given
time or space
Poissoinpdf(,x)
P( x) 
(a) x = 0; poissonpdf(0.1870,0) = 0.829
(b) x = 1; poissonpdf(0.1870,1) = 0.155
(c) x > 1 = 1 – (P(0) + P(1)) = 0.016
 x e 
x!
36 yrs = 13140 days
Chapter 4 Practice Problems
25. A newspaper finds that the mean number of typographical errors per page
is four. Find the probability that (a) exactly three typographical errors will be
found on a page, (b) at most three typographical errors will be found on a page,
and (c) more than three typographical errors will be found on a page.
x = # occurrences in a
given time or space
= average # of
occurrences in a given
time or space
P( x) 
 x e 
x!
Poissoinpdf(,x)
(a)  = 4, x = 3; P(3) poissonpdf(4,3) = 0.195
(b) P(x  3) = P(0) + P(1) + P(2) + P(3) = .0183 + .0733 +
0.1465 + 0.1954 = 0.433
(c) P(x > 3) = 1 – P( x  3) = 1 – 0.433 = 0.567
Chapter 3 Practice Problems
26. A single six-sided die is rolled. Find the probability of rolling a number less than 3.
A) 0.25
B) 0.333
1) B
C) 0.1
D) 0.5
2 correct answers/6 possible answers =0.33
27. In a survey of college students, 880 said that they have cheated on an exam
and 1721 said that they have not. If one college student is selected at random, find
the probability that the student has cheated on an exam.
A) 1721/2601
4) D
B) 2601/1721
C) 2601/880
D) 880/2601
880 correct answers/(1721+880) possible answers = 880/2601
Chapter 3 Practice Problems
28. The distribution of blood types for 100 Americans is listed in the table. If one
donor is selected at random, find the probability of selecting a person with blood
type A+ or A-.
Blood Type O+ O- A+ A- B+ B- AB+ ABNumber
37
6 34 6 10
2
4
1
A) 0.34
5) C
B) 0.06
C) 0.4
D) 0.02
(34 +6) correct answers/100 possible answers = 40/100 = 0.4
29. A card is picked at random from a standard deck of 52 playing cards. Find the
odds that it is not a heart.
A) 1:3
10) D
B) 1:4
C) 4:1
D) 3:1
1 suit of correct answers/4 suits of possible
answers = ¼. To not be a heart means that we
have 4/4 – ¼ = ¾ . There are a 3:1 odds that a
heart will not be chosen randomly.
Chapter 3 Practice Problems
29. Classify the events as dependent or independent. Events A and B where
P(A) = 0.6, P(B) = 0.3, and P(A and B) = 0.17
A) independent
12) B
B) dependent
The results of P(A and B) = 0.17 indicates that the event A
and the event B are interrelated and so they are dependent.
30. Find the probability of answering two true or false questions correctly if random
guesses are made. Only one of the choices is correct.
A) 0.25
15) A
B) 0.75
C) 0.1
D) 0.5
1 correct answer/4 possible answers =0.25
Chapter 3 Practice Problems
30. Find the probability of getting four consecutive aces when four cards are drawn
without replacement from a standard deck of 52 playing cards.
30. P(4-Aces) = (4/52)(3/51)(2/50)(1/49 )= 0.00000369
31. A multiple-choice test has five questions, each with five choices for the answer.
Only one of the choices is correct. You randomly guess the answer to each question.
What is the probability that you do not answer any of the questions correctly?
31. P(all five questions answers incorrect) = (4/5)(4/5)(4/5)(4/5)(4/5) = 0.32768
32. The probability it will rain is 40% each day over a three-day period. What is the
probability it will rain at least one of the three days?
32. P(rain at least one day) = 1 - P(no rain all three days)= 1 - (0.60)(0.60)(0.60) = 0.7845
Chapter 3 Practice Problems
32. A card is selected at random from a standard deck. Find each probability.
13 4 1 16
 

 0.308
(a) Randomly selecting a diamond or a 7.
_________________________
52 52 52 52
(b) Randomly selecting a red suit or a queen.
(c) Randomly selecting a 3 or a face card
26 4
2 28
 

 0.538
52 52 52 52
_________________________
4 12 16


 0.308
52
52
52
_________________________
33. You roll one die. Find each probability.
(a) Rolling a 6 or a number greater than 4.
1 2 1 2
    0.333
6 6 6 6
_________________________
(b) Rolling a number less than 5 or an odd number.
4 3 2 5
    0.833
_________________________
6 6 6 6
(c) Rolling a 3 or an even number.
1 3 4
   0.667
6 6 6
_________________________
Chapter 3 Practice Problems
34. Determine the probability that an 8-sided die, numbered 1-8 is rolled and the results
of the roll is an even number or a number that is greater than 6.
4 2 1 5
    0.625
___________________________________
8 8 8 8
Chapter 3 Practice Problems
35.
Gender
Male
Female
Total
Level
Associates
260
405
665
Of
Bachelor’s
595
804
1399
Degree
Master’s
230
329
559
Doctorate
25
23
48
Total
1110
1561
2671
(a) earned a master’s degree or is a female
(b) earned an associate degree and is a male
559 1561 329


 0.671
2671
2671
2671
_________________
260
 0.097
2671
_________________
804
 0.575
1399
(c) is a female given that the person earned a bachelor’s degree_________________
Chapter 3 Practice Problems
36. Using a standard 52-card deck for each situation, determine the probability.
(a) Drawing a five and a spade.
(b) Drawing a five of spades.
4 1
4
 
 0.019
52 4 208
1
 0.019
52
(c) Drawing a spade and then without replacement, drawing club.
13 13 169
 
 0.064
52 51 2652
(d) Drawing a 3 of clubs and then without replacement, drawing another club.
1 12
12
 
 0.005
52 51 2652
(e) Drawing a 4 of diamonds or a heart on the first draw.
1 13 14


 0.269
52 52 52
Chapter 3 Practice Problems
37. How many ways can a jury of five men and three women be selected from twelve
men and ten women?
(12C5)(10C3) = 95,040
38. How many different permutations of the letters in the word PROBABILITY are there?
11!/(2!2!) = 9,979,200
39. A student must answer six questions on an exam that contains twelve questions.
a) How many ways can the student do this?
b) How many ways are there if the student must answer the first and last question?
(a) 12C6 = 924; (b) 10C4 = 210
Chapter 3 Practice Problems
40. In the California State lottery, you must select six numbers from fifty-two numbers to
win the big prize. The numbers do not have to be in a particular order. What is the
probability that you will win the big prize if you buy one ticket?
1
52C6
= 1/20,358,520 = 0.0000000491
41. From a group of 40 people, a jury of 12 people is selected. In how many different
ways can a jury of 12 people be selected?
5,586,853,480
40 C12 
40!
 5,586,853,480
(40  12)!(12!)
Chapter 3 Practice Problems
42. An area code consists of three digits. How many area codes are possible if
(a) there are no restrictions and (b) if the first digit can not be 1 or 0. (c) What is
the probability of selecting an area code at random that ends in an odd number
if the first digit cannot be a 1 or a 0?
42. (a) 10x10x10= 1000 (b) 10x10x8= 800
(c) 0.5
43. A shipment of 10 microwave ovens contains two defective units. In how many
ways can a restaurant buy three of these units and receive (a) no defective units,
(b) one defective unit, and (c) at least two non-defective units? (d) What is the
probability of the restaurant buying at least two defective units?
43. (a) 56
(b) 56
(c) 112
(d) 0.067
(a) (8C3 )(2C0 ) = (56 )(1 )= 56
(b) (8C2 )(2C1 ) = (28 )(2 )= 56
(c) At least two good units one or fewer defective units. 56 + 56 = 112
(d) Pat least 2 defective units =
8
C1  2 C2 8  1

 0.067
120
10 C3