# Practice Questions for Business Statistics

## Contents:

40-1 Chance Process

## Questions:

25-2

```    Q:  Which of the following pairs of events are mutually exclusive?
a.  A:  the odd numbers;          B:  the number 5
b.  A:  the even numbers;         B:  the numbers greater than 10
c.  A:  the numbers less than 5;  B:  all negative numbers
d.  A:  the numbers above 100;    B:  the numbers less than -200
e.  A:  negative numbers;         B:  odd numbers
```

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27-1

```    Q:  One card is drawn from a standard 52 card deck.  In describing the
occurrence of two possible events, an Ace and a King, these two
events are said to be:

(a)  independent
(b)  mutually exclusive
(c)  random variables
(d)  randomly independent.
```

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34-1

```    Q:  Suppose a certain opthalmic trait is associated with eye color.
300 randomly selected individuals are studied with results
as follows:

EYE COLOR
TRAIT | Blue | Brown | Other | Total
____________________________________
Yes   |  70  |   30  |   20  |  120
____________________________________
No    |  20  |  110  |   50  |  180
____________________________________
Total |  90  |  140  |   70  |  300

A.  What is the probability that a person has blue eyes?

B.  What would you expect to be the value P(having the trait and
blue eyes) if eye color and trait status were independent?

C.  Which of the following expressions describes the relation-
ship between the events A = a person has brown eyes and
B = a person has blue eyes? (circle the correct answer)

i. independent      ii. exhaustive
iii. simple           iv. mutually exclusive
```

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37-2

```    Q:  True or False?  If False, correct it.

The law of large numbers makes it possible to predict long run relative
frequencies but not particular chance events.
```

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38-2

```    Q:  Define the complement to the event A.
```

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40-1

```    Q:  Define the following term and give an example of its use.
Your example should not be one given in class or in a handout.
Chance Process
```

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44-2

```    Q:  Which of the following is NOT a possible probability?

a.  25/100
b.  1.25
c.  1
d.  0
```

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71-1

```    Q:  A sample of 1000 persons screened for a certain disease is distributed
according to height and disease status resulting from a clinical exam
as follows:

DISEASE STATUS

None     Mild     Moderate     Severe    Totals
|-------|---------|------------|----------|
Tall  |  122  |    78   |    139     |    61    |  400
|-------|---------|------------|----------|
HEIGHT  Medium  |   74  |    51   |     90     |    35    |  250
|-------|---------|------------|----------|
Short  |  104  |    71   |    121     |    54    |  350
|-------|---------|------------|----------|
300      200        350         150      1000

What would you estimate from the above table to be the probability
of being medium or short in height and having moderate or severe
disease status?

a.  600/1000 * 500/1000      d.  300/600
b.  300/500                  e.  800/1000
c.  300/1000
```

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533-2

```    Q:  Among twenty-five articles, nine are defective, six having only minor
defects and three having major defects.   Determine  the  probability
that  an  article  selected at random has major defects given that it
has defects.

a.  1/3
b.  .25
c.  .24
d.  .08
```

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534-1

```    Q:  The checking accounts of Save-More Bank are categorized by age of
account and balance in account.  We are going to select an account at
random from this group of 2000 accounts.

Balance |            |            |
Age of Account       |  \$0 - \$99  |  \$100-\$499 |  \$500 or more
---------------------------------------------------------------
less than 3 years |    700     |     100    |      400
3 or more years   |    200     |     400    |      200
---------------------------------------------------------------

i)  Then P(\$500 or more0 - 3 years) =

a) 6/7    b) 2/3    c) 1/3    d) 1/5    e) none of these

ii)  Then P[(\$0 - \$99) or (3 years or more)] =

a) 9/10   b) 9/20   c) 3/4    d) 2/5    e) none of these

iii)  Then P(\$100 or more) =

a) 1/4    b) 3/20   c) 11/20  d) 3/10   e) none of these

iv)  What is the conditional probability that the account has a
balance under \$100, given that it is less than 3 years old?

a) 7/9    b) 9/20   c) 3/4    d) 7/12   e) none of these

v)  Are age of account and balance in account independent at Save-More
Bank?  Why or why not?

vi)  Suppose fourteen accounts are drawn at random from this bank.  Let
G be the event: "At least five accounts are less than 3 years old".
State G', the complement of G.
```

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536-1

```    Q:  The depositors at Save-More Bank are categorized by age and sex.  We
are going to select an individual at random from this group of 2000
depositors.

Sex
Age          |    Male    |  Female
-------------------------------------
30 or less   |     800    |    600
31 or more   |     400    |    200
-------------------------------------

i)  Then P(Female30 or less) =
a) 2/5    b) 3/4    c) 3/7    d) 3/10    e) none of these

ii)  Then P[Male or (31 or more)] =
a) 1/5    b) 3/10    c) 1/2    d) 7/10    e) none of these

iii)  Then P(Female) =
a) 3/10    b) 2/5    c) 3/5    d) 2/3    e) none of these

iv)  What is the conditional probability that the depositor drawn is
30 or less, given that he is a male?
a)  2/3    b) 7/10    c) 4/7    d) 2/5    e) none of these

v)  Are age of depositor and sex of depositor independent at Save-More
Bank?  Why or why not?
```

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538-1

```    Q:  The Green Bay Packers and Chicago Bears are playing their annual charity
game.  It is third down, four yards to go  for  the Packers on their own
45 yard line.  Based on previous performances by  the Packers,  the next
play will see one of the following occurrences,  with  probabilities  as
indicated:

P (pass incomplete)                  = .25
P (pass complete for first down)     = .15
P (pass complete, no first down)     = .05
P (interception)                     = .03
P (run for first down)               = .20
P (busted play, quarterback sacked)  = .25
P (whistle is blown before play)     = .05

Find the following probabilities for the next play:

a)  P (Packers score a first down)
b)  P (pass play is tried)
c)  P (Packers score a first downpass play is tried)
```

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539-1

```    Q:  400 adult males with angina pectoris are classified by age and weight
as follows:

|          Weight in Pounds           |
Age (years)  | 130-149 | 150-169 | 170-189 | >=190 | Total
-----------------------------------------------------------
30-39    |   10    |   20    |   20    |  40   |   90
40-49    |   10    |   15    |   50    |  70   |  145
50-59    |    5    |   15    |   50    |  40   |  110
60-69    |    5    |   10    |   15    |  25   |   55
-----------------------------------------------------------
|   30    |   60    |  135    | 175   |  400

Using the table, find for a randomly selected individual from this
population the probability that he or she:

a)  Is in the age interval 40-49
b)  Is in the age interval 40-49 and weighs 170-189 lbs
c)  Is in the age interval 40-49 or 60-69
d)  Is in the age interval 40-49 or 60-69 and weighs 150-169 lbs
e)  Is in the age interval 40-49 given a weight between 150-169 lbs
f)  Weighs less than 170 lbs
g)  Weighs less than 170 lbs and is less than 50 years
h)  Weighs less than 170 lbs given that he is less than 50 years
```

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540-1

```    Q:  Each member of a sample of 3,750 women aged 30-39 was measured for
unaided distance vision in both right and left eyes.  The results are
presented in the following table:

|             LEFT EYE             |
RIGHT EYE   |  Highest  Second  Third  Lowest  |  Total
---------------------------------------------------------
Highest grade |    750      130     60      35   |   975
Second grade  |    125      775    210      40   |  1150
Third grade   |     60      180    885     100   |  1225
Lowest grade  |     20       40     90     250   |   400
---------------------------------------------------------
Total         |    955     1125   1245     425   |  3750

For a randomly selected person from the population sampled above, what
is your estimate of the probability that:

a)  the left eye will fall into the 3rd grade of unaided distance vision
b)  the left eye will have the highest grade given that the right eye
c)  the right eye will have the highest grade and the left eye will have
```

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541-1

```    Q:  The following two-way table shows the frequencies of occurrence of a
hypothetical exposure and disease in a group of 1000 people.

Disease
Present     Absent  |
Present      75          325   |  400
Exposure  Absent       25          575   |  600
---------------------------
100          900     1000

a.  What is the probability of exposure in the group?
b.  What is the joint probability of both exposure and disease
being present in the group?
c.  Compute the probability of disease being present conditional
on the presence of exposure and conditional on the absence
of exposure.
```

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542-1

```    Q:  An epidemiologist feels that railroads have something to do with the
development of a new disease because the probability of a person's
living within a mile of railroad tracks, given that he has the disease,
is .80.  Do you agree with him?  Why or why not?
```

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545-1

```    Q:  Among twenty-five articles eight are defective, six having only
minor defects and two having major defects.  Determine the pro-
bability that an article selected at random has major defects
given that it has defects.

(a)  .08              (c)  1/3
(b)  .25              (d)  .24
```

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546-1

```    Q:  A dormitory on campus houses 200 students.  120 are male, 50 are
upper division students, and 40 are upper division male students.
A student is selected at random.

The probability of selecting a lower division student, given the
student is a female, is:

(a)  7/8            (d)  7/20
(b)  7/15           (e)  1/4
(c)  2/5
```

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549-1

```    Q:  A local trade union consists of plumbers and electricians.
Classified according to rank:

Apprentice    Journeyman      Master
------------------------------------
Plumbers          25      |     20       |    30      75
------------------------------------
Electricians      15      |     40       |    20      75
-------------------------------------
40            60            50

A member of the union is selected at random.  Given that the
person selected is a plumber, the probability that he is a
journeyman is:

a.  1/2
b.  1/3
c.  4/15
d.  2/15
e.  none of these.
```

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550-1

```    Q:  In an effort to get at the source of an outbreak of Legionnaire's
disease at the 197? APHA convention, a team of medical detectives
(i.e. epidemiologists) carried out a case-control study involving
all 50 cases and a sample of 200 non-cases out of the 4000 persons
attending the convention.  Among the results, it was found that 40%
of the cases went to a cocktail party given by a large drug company
on the second night of the convention, whereas 10% of the controls
attended the same party.  Which of the following statements is
appropriate for describing the 40% of cases who went to the party?
(C = case, P = attended party)

a.  Pr(C|P) = .40      b.  Pr(P|C') = .40      c.  Pr(C|P') = .40
d.  Pr(P'|C) = .40     e.  none of these
```

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553-2

```    Q:  Suppose a loaded die has the following model:

Face          1       2       3        4       5       6

Probability  0.3     0.1     0.1     0.1     0.1     0.3

If this die is thrown and the top face shows an odd number,

a.  What is the probability that the die shows a four?
b.  What is the probability that the die shows a 1?
```

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558-1

```    Q:  There are two urns marked H and T.  Urn H contains 2 red marbles and
1 blue marble.  Urn T contains 1 red and 2 blue marbles.  A coin is
to be tossed.  If it lands heads, a marble is drawn from Urn H.  If
it lands tails a marble is drawn from Urn T.  Find the following pro-
babilities:

-------------------
|    H   |    T   |
-----------------------------
R   |   2/6  |   1/6  |   1/2
-----------------------------
B   |   1/6  |   2/6  |   1/2
-----------------------------
|   1/2  |   1/2  |    1
-------------------------

a.  P(heads and red)      b.  P(tails)          c.  P(red)
```

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560-1

```    Q:  A sample of 2000 individuals is distributed according to eye color
and the presence or absence of a certain opthalmic trait as follows:

|          EYE COLOR           |
TRAIT   |  Blue      Brown      Other  |
----------------------------------------------
Yes     |  400        270        130   |   800
No      |  200        650        350   |  1200
----------------------------------------------
Total   |  600        920        480   |  2000

In a random selection of an individual from the study population,
what is your estimate of the probability that:

a.  the person has blue eyes? _______________
b.  the trait is present and the person has brown eyes? ____________
c.  the person has neither brown nor blue eyes given that the trait
is absent? _______________
d.  the person has neither brown nor blue eyes and the trait is pre-
sent? _______________
e.  the person does not have brown eyes? _______________
f.  the person has blue eyes or has neither blue nor brown eyes?
_____________
g.  the person does not have the trait or does not have brown eyes?
_______________
```

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1885-1

```    Q:  How may the standard deviation be expressed?

a.  A point on a z-score scale.
b.  A distance on a z-score scale.
c.  An index on a squared numerical scale.
d.  (a) and (c) are both correct.
e.  (b) and (c) are both correct.
```

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25-2

```    A:  d.  A:  the numbers above 100;   B:  the numbers less than -200
```

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27-1

```    A:  (b)  mutually exclusive.
```

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34-1

```    A:  A.  90/300 = .3
B.  (120/300)*(90/300) = .12
C.  iv.  mutually exhaustive
```

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37-2

```    A:  True.
```

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38-2

```    A:  The complement of A is the set of elements which do not belong to A.
```

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40-1

```    A:  Definition:  A process producing results or outcomes where the next
outcome cannot be specified in advance, but where the long
term chances of that outcome can be determined.
Example:     If we were to regard dropping a baited line into water
suspected of harboring flounder as a random process, we
would be taking the position that it's not possible to
accurately forecast if the next dropping of a line will
return a fish, but that if we repeatedly drop a properly
baited line it will result in a catch some percentage of
the time.
```

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44-2

```    A:  b.  1.25

0 <= Probability <= 1.0
```

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71-1

```    A:  c.  300/1000

(90 + 35 + 121 + 54)/1000 = 300/1000
```

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533-2

```    A:  a.  1/3

P[MD/D] = (3/25)/(9/25)
= 3/9
= 1/3
```

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534-1

```    A:  i)  c) 1/3

P(\$500 or more0 - 3 years) = 400/1200 = 1/3

ii)  c) 3/4

P((\$0 - \$99) or (3 years or more))
= P(\$0 - \$99) + P(3 years or more) - P((\$0 - \$99) and (3 years))
= 900/2000 + 800/2000 - 200/2000
= 3/4

iii)  c) 11/20

P(\$100 or more) = P(\$100 - \$499) + P(\$500 or more)
= 500/2000 + 600/2000
= 11/20

iv) d) 7/12

P(\$0 - \$990 - 3 years) = 700/1200
= 7/12

v)  The two variables are not independent because
P(0 - 3 years\$0 - \$99) =/=
P(0 - 3 years\$100 - \$499) =/=
P(0 - 3 years\$500 or more)

vi)  G': "At least 10 accounts are three years old or older"
or
"less than five accounts are less than three years old"
```

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536-1

```    A:  i)  c) 3/7

P(Female30 or less) = 600/1400 = 3/7

ii)  d) 7/10

P[Male or (31 or more)] = P(Male) + P(31 or more)
- P(Male and 31 or more)
= (1200/2000) + (600/2000) - (400/2000)
= 1400/2000
= 7/10

iii)  b) 2/5

P(Female) = 800/2000 = 2/5

iv)  a) 2/3

P(30 or lessMale) = 800/1200 = 2/3

v)  They are not independent because
P(30 or lessMale) =/= P(30 or lessFemale)
```

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538-1

```    A:  a)  .15 + .20 = .35
b)  .25 + .15 + .05 + .03 = .48
c)  .15/.48 = .3125
```

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539-1

```    A:  a)  .3625 = 145/400
b)  .1250 = 50/400
c)  .5000 = 145/400 + 55/400
d)  .0625 = 15/400 + 10/400
e)  .2500 = 15/60
f)  .2250 = 30/400 + 60/400
g)  .1375 = 10/400  + 10/400 + 20/400 + 15/400
h)  .2340 = (10+10+20+15)/(145+90)
```

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540-1

```    A:  a)  1245/3750 = .3320
b)  20/400 = .05
c)  35/3750 = .0093
```

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541-1

```    A:  a.  P(exposure) = 400/1000 = .40
b.  P(exposure and disease) = 75/1000 = .075
c.  P(diseaseexposure present) = 75/400 = .1875
P(diseaseexposure absent)  = 25/600 = .0417
```

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542-1

```    A:  You don't know whether P(living near tracksdisease) is equal to
P(living near tracks), therefore you cannot evaluate whether the two
events are independent or not.
```

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545-1

```    A:  (b)  .25

P(MDD) = P(MD and D)/P(D)
= (2/25)/(8/25)
= .25
```

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546-1

```    A:  (a)  7/8

Given the above information, one can construct the following table:

Lower    Upper

Male    |    80    |   40    |  120
-------------------------------------
Female  |    70    |   10    |   80
-------------------------------------
|   150    |   50    |  200

P(LowerFemale) = 70/80
```

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549-1

```    A:  c.  4/15

20/75 = 4/15
```

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550-1

```    A:  e.  none of these

Pr(PC) = .40 is the correct notation.
```

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553-2

```    A:  a.  P(4|odd number) = 0

b.  P(1|odd number) = P(1 and odd number)/P(odd number)
= .3/(.3 + .1 + .1)
= .6
```

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558-1

```    A:  (a)  P(heads, red) = (1/2)*(2/3) = 1/3

(b)  P(tails) = 1/2

(c)  P(red) = ((1/2)*(2/3) + (1/2)*(1/3)) = 1/2

(d)  P(blue) = ((1/2)*(1/3) + (1/2)*(2/3)) = 1/2

= (1/3)/(1/2) = (1/3)*(2/1) = 2/3
```

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560-1

```    A:  a.  600/2000 = 3/10

b.  270/2000 = 27/200

c.  350/1200 = 7/24

d.  130/2000 = 13/200

e.  (600 + 480)/2000 = 27/50

f.  P(Blue union Other) = P(Blue) + P(Other) - P(Blue and Other)
= 600/2000 + 480/2000 - 0
= 1080/2000 = 27/50

g.  P(No trait union (Blue or Other))
= 1200/2000 + 1080/2000 - 550/2000
= 1730/2000 = 173/200
```

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1885-1

```    A:  b.  A distance on a z-score scale.
```

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## Identification:

25-2

```Item is still being reviewed
Multiple Choice
BASICTERMS/PROB
PROBABILITY

T= 2    Comprehension
D= 2    General
```

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27-1

```Item is still being reviewed
Multiple Choice
BASICTERMS/PROB   EVENTS
PROBABILITY

T= 2    Comprehension
D= 2    General
```

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34-1

```Based upon item submitted by D. Kleinbaum - Univ of North Carolina
INDEPENDENT       OTHER/EVENTS      BASICTERMS/PROB
SIMPLEPROBABILIT  EVENTS            PROBABILITY

T=10    Comprehension
D= 3    General             Biological Sciences
***Multiple Parts***
```

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37-2

```Based upon item submitted by W. J. Hall - Univ. of Rochester
True/False
BASICTERMS/PROB
PROBABILITY

T= 2    Comprehension
D= 3    General
```

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38-2

```Based upon item submitted by S. Selvin - UC Berkeley
Definition
OTHER/EVENTS      BASICTERMS/PROB
EVENTS            PROBABILITY

T= 2    Comprehension
D= 1    General
```

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40-1

```Based upon item submitted by J. Warren - UNH
Definition
BASICTERMS/PROB   EVENTS
I650I             PROBABILITY

T= 5    Comprehension
D= 2    General
```

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44-2

```Item is still being reviewed
Multiple Choice
SIMPLEPROBABILIT  PROBMODELS
PROBABILITY

T= 2    Comprehension
D= 1    General             Education
```

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71-1

```Item is still being reviewed
Multiple Choice
JOINTDISTRIBUTIO  CONTINGENCYTABLE
PROBDISTRIBUTION  PROBABILITY       CHISQUARE
NONPARAMETRIC     STATISTICS

T= 2    Computation
D= 3    Biological Sciences General
```

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533-2

```Based upon item submitted by H. B. Christensen - BYU
Multiple Choice
CONDITIONALPROBA  NONBAYESIAN
PROBABILITY

T= 2    Computation     Application
```

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534-1

```Item is still being reviewed
Multiple Choice
INDEPENDENT       CONDITIONALPROBA  JOINTPROBABILITY
EVENTS            PROBABILITY

T=10    Computation     Comprehension
D= 3    General
***Multiple Parts***
```

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536-1

```Item is still being reviewed
Multiple Choice
INDEPENDENT       CONDITIONALPROBA  JOINTPROBABILITY
EVENTS            PROBABILITY

T=10    Computation     Comprehension
D= 4    General
```

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538-1

```Based upon item submitted by R. H. Locher - Marquette
SIMPLEPROBABILIT  CONDITIONALPROBA
PROBABILITY

T= 5    Computation
D= 3    General
***Multiple Parts***
```

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539-1

```Based upon item submitted by D. Kleinbaum - Univ of North Carolina
SIMPLEPROBABILIT  CONDITIONALPROBA
PROBABILITY

T=10    Computation
D= 3    General             Biological Sciences
***Calculator Necessary***
```

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540-1

```Based upon item submitted by D. Kleinbaum - Univ of North Carolina
SIMPLEPROBABILIT  CONDITIONALPROBA
PROBABILITY

T=10    Computation
D= 4    General
***Calculator Necessary***
```

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541-1

```Based upon item submitted by A. Matthews - U. Mass.
CONDITIONALPROBA  SIMPLEPROBABILIT  JOINTPROBABILITY
PROBABILITY

T= 5    Computation
D= 3    General             Biological Sciences
***Multiple Parts***
```

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542-1

```Item is still being reviewed
EVENTS            CONDITIONALPROBA
PROBABILITY

T= 5    Comprehension
D= 5    General             Biological Sciences
```

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545-1

```Based upon item submitted by H. B. Christensen - BYU
Multiple Choice
NONBAYESIAN
CONDITIONALPROBA  PROBABILITY

T= 5    Computation
```

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546-1

```Based upon item submitted by F. J. Samaniego - UC Davis
Multiple Choice
NONBAYESIAN
CONDITIONALPROBA  PROBABILITY

T= 5    Computation
D= 2    General
```

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549-1

```Based upon item submitted by F. J. Samaniego - UC Davis
Multiple Choice
SIMPLEPROBABILIT  NONBAYESIAN
PROBABILITY       CONDITIONALPROBA

T= 2    Computation
D= 2    General
```

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550-1

```Based upon item submitted by D. Kleinbaum - Univ of North Carolina
Multiple Choice
NONBAYESIAN
CONDITIONALPROBA  PROBABILITY

T= 2    Comprehension
D= 3    Biological Sciences
```

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553-2

```Item is still being reviewed
NONBAYESIAN
CONDITIONALPROBA  PROBABILITY

T= 2    Application
D= 3    General
***Multiple Parts***
```

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558-1

```Based upon item submitted by W. Geeslin - UNH
NONBAYESIAN       SIMPLEPROBABILIT
CONDITIONALPROBA  PROBABILITY

T= 5    Computation     Application
D= 3    General
***Multiple Parts***
```

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560-1

```Based upon item submitted by D. Kleinbaum - Univ of North Carolina
SIMPLEPROBABILIT  NONBAYESIAN
PROBABILITY       CONDITIONALPROBA

T=10    Computation
D= 2    General             Biological Sciences
***Multiple Parts***
```

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1885-1

```Item is still being reviewed
Multiple Choice
STANDARDDEVIATIO
DESCRSTAT/P       PARAMETRIC        STATISTICS

T= 2    Comprehension
D= 2    General
```

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