MCQ
on Introduction to statistics Variable as attribute of an entity
Highlighted option is the answer
1. The
specific statistical methods that can be used to summarize or to describe a
collection of data is called:
a) Descriptive statistics
b) Inferential
statistics
c) Analytical
statistics
d) All
of the above
2. The
need for inferential statistical methods derives from the need for
______________.
a)
Population
b) Association
c) Sampling
d) Probability
3. A population, in statistical terms, is the totality of
things under consideration. It is the collection of all values of the
_________________ that is under study.
a)
Instance
b) Variable
c) Amount
d) Measure
4. Non-sampling
errors are introduced due to technically faulty observations or during the
______________________ of data.
a) Processing
b) Analysis
c) Sequencing
d) Collection
5. Sampling
is simply a process of learning about the __________________ on the basis of a
sample drawn from it.
a) Census
b) Population
c) Group
d) Area
6. Numerical
facts are usually subjected to statistical analysis with a view to helping a
decisionmaker make wise decisions in the face of ___________________.
a) Interpreting
b) Uncertainty
c) Summarizing
d) Organizing
7. In
statistics, ___________________________ classification includes data according
to the time period in which the items under consideration occurred.
a) Chronological
b) Alphabetical
c) Geographical
d) Topological
8. Data
is simply the numerical results of any scientific__________________.
a) Analysis
b) Researches
c) Observation
d) Measurement
9. The
________________ process would be required to ensure that the data is complete
and as required.
a) Tabulation
b) Analysis
c) Editing
d) Ordering
10. A
sample is a portion of the ________________ population that is considered for
study and analysis.
a) Selected
b) Total
c) Fixed
d) Random
11. The
method of sampling, in which the choice of sample items depends exclusively on
the judgement of the investigator is termed as ________________________.
a) Convenience
sampling
b) Quota
sampling
c) Systematic
sampling
d) Judgement sampling
12. Both
the sampling as well as the non-sampling errors must be reduced to a minimum in
order to get as representative a sample of the ___________________ as possible.
a) Group
b) Region
c) Population
d) Universe
13. The
larger the size of the population, the ___________________ should be the sample
size.
a) Smaller
b) Larger
c) Accurate
d) Fixed
14. When
the data is to be processed by computers, then it must be coded and converted
into the ____________________ ___________________.
a) English
language
b) Regional
language
c) Statistical
language
d) Computer language
15. A
variable is any characteristic which can assume ____________________ values.
a) Different
b) Similar
c) Fixed
d) Assumed
16. The
basic objective of a sample is to draw ____________________ about the
population from which such sample is drawn.
a) Conclusion
b) Characteristics
c) Inferences
d) Parameters
17. In
___________________ type of classification, the data is grouped together
according to some distinguished characteristic or attribute, such as religion,
sex, age, national origin, and so on.
a) Quantitative
b) Chronological
c) Qualitative
d) All
of the above
18. A
_____________________ variable is a variable whose values can theoretically
take on an infinite number of values within a given range of values.
a) Continuous
b) Discrete
c) Random
d) Both
(a) and (b)
19. A
perfect random number table would be one in which every digit has been entered
_______________.
a) Chronologically
b) Sequentially
c) Randomly
d) Arbitrarily
20. The
_________________ random variables yield categorical responses so that the
responses fit into one category or another.
a) Quantitative
b) Discrete
c) Continuous
d) Qualitative
21. For
a sample to be truly representative of the population, it must truly
be________________.
a) Fixed
b) Random
c) Specific
d) Casual
22. A
______________ ______________ is a phenomenon of interest in which the observed
outcomes of an activity are entirely by chance, are absolutely unpredictable
and may differ from response to response.
a) Discrete
variable
b) Continuous
variable
c) Random variable
d) All
of the above
23. By
definition of randomness, each ________________ ______________ has the same
chance of being considered.
a) Possible entity
b) Probable
entity
c) Random
entity
d) Observed
entity
24. Before
any procedures for _____________ _________________ are established, the purpose
and the scope of the study must be clearly specified.
a) Data
analysis
b) Data
tabulation
c) Data collection
d) Data
selection
25. Adequacy
of data is to be judged in the light of the requirements of the survey and the
geographical areas covered by the __________________ data.
a) Collected
b) Available
c) Organized
d) Tabulated
26. If
the sample is truly representative of the population, then the characteristics
of the sample can be considered to be the same as those of the _______________
population.
a) Fixed
b) Selected
c) Random
d) Entire
27. Statistical
inference deals with methods of inferring or drawing ___________________ about
the characteristics of the population based upon the results of the sample
taken from the same population.
a) Details
b) Decisions
c) Conclusions
d) Samples
28. If
the sample size is too small, it may not _______________ represent the
population or the universe as it is known, thus leading to incorrect
inferences.
a) Appropriately
b) Reliably
c) Homogeneously
d) Heterogeneously
29. Editing
would also help eliminate inconsistencies or obvious errors due to
_______________ treatment.
a) Characteristic
b) Arithmetical
c) Calculation
d) Tabulation
30. When
an investigator uses the data which has already been collected by others, such
data is called _______________ _________.
a) Primary
data
b) Collected
data
c) Processed
data
d) Secondary data
31. In
the case of the questionnaire method of gathering data, it should be made
certain that all the questions have been _____________________.
a) Read
b) Interpreted
c) Answered
d) All
of the above
32. _____________________
provides various types of statistical information of either qualitative or
quantitative nature.
a) Sampling
b) Tabulation
c) Observation
d) Editing
33. In
statistics, ____________________classification groups the data according to
locational differences among the items.
a) Chronological
b) Geographical
c) Regional
d) Alphabetical
34. The
degree of randomness of selection would depend upon the process of selecting the
items from the ________________________.
a) Population
b) Region
c) Sample
d) Data
35. A
_____________________ sample is obtained by selecting convenient population
units
a) Random
b) Quota
c) Stratified
d) Convenience
36. A
__________________ sample is formed by selecting one unit at random and then
selecting additional units at evenly spaced intervals until the sample has been
formed.
a) Stratified
b) Systematic
c) Judgement
d) Random
37. The
sampling errors arise due to drawing faulty inferences about the
__________________ based upon the results of the samples.
a) Sample
b) Survey
c) Population
d) Census
38. A
summary measure that describes any given characteristic of the population is
known as a __________________.
a) Parameter
b) Information
c) Inference
d) Statistics
39. ________________
means separating items according to similar characteristics and grouping them
into various classes.
a) Tabulation
b) Editing
c) Separation
d) Classification
40. _____________________
is one which is collected by the investigator himself for the purpose of a
specific inquiry or study.
a) Secondary
data
b) Primary data
c) Statistical
data
d) Published
data
UNIT 2 ANALYSIS OF STATISTICAL DATA
1. In
chronological classification, the data is classified on the basis of:
a) Time
b) Money
c) Location
d) Quality
2. The
classification of data according to location is what classification:
a)
Chronological
b)
Quantitative
c)
Qualitative
d)
Geographical
3. The
magnitude of the class is the:
a) The
product of lower limit and upper limit
b) The
sum of lower limit and upper limit
c) The difference of upper limit and lower limit
d) None
of these
4. A
function very similar to that of sorting letters in a post office is:
a) Mean
b) Standard
deviation
c) Classification
d) Mean
deviation
5. The
value lying half way between the upper limit and lower limit of the class is:
a) Class
interval
b) Mid point
c) Frequency
d) None
of the above
6. The
classes in which the lower limit or the upper limit is not specified are known
as:
a) Open end classes
b) Close
end classes
c) Inclusive
classes
d) Exclusive
classes
7. Classes
in which upper limits are excluded from the respective classes and are included
in the immediate next class are:
a) Open
end classes
b) Close
end classes
c) Inclusive
classes
d) Exclusive classes
8. If
the class mid points in a frequency distribution of age of a group of persons
are 25, 32, 39, 46, 53 and 60. The size of class interval is:
a) 5
b) 7
c) 8
d) 6
9. The
number of observations in a particular class is called:
a) Width
of the class
b) Class
mark
c) Frequency
d) None
of the above
10. If
the mid points of the classes are 16, 24, 32, 40, and so on, then the magnitude
of the class interval is:
a) 8
b) 9
c) 7
d) 6
11. The first step in tabulation is:
a) Foot
note
b) Source
note
c) Captions
d) Classification
12. A systematic arrangement of data in rows
and columns is: a) Table
b) Tabulation
c) Body
d) All
the above
13. The
numerical information in a statistical table is called the: a) Table
b) Foot
note
c) Source
note
d) Body
14. In a statistical table the row headings are referred to
as:
a) Source
note
b) Captions
c) Stubs
d) Body
15. In the statistical table column headings are called:
a) Stubs
b) Captions
c) Source
note
d) None
of these
16. If
the class mid points in a frequency distribution of a group of persons are:
125, 132, 139,
146, 153, 160, 167,
174, 181 pounds, then the size of the class is: a)
6
b) 8
c) 7
d) 9
17. The
different types of samplings are:
a) Probability
b) Judgement
c) Mixed
d) All the above
18. Two
dimensional diagrams used in surface diagrams are:
a) Squares
b) Pie
diagrams
c) Circles
d) All the above
19. One
dimensional diagram is:
a) Line diagram
b) Rectangles
c) Cubes
d) Squares
20. Type
of bar diagram is:
a) Pictogram
b) Sub divided diagram
c) Line
diagrams
d) Pie
diagram
21. The most
commonly used device of presenting business
and economic data is: a) Pie diagrams
b) Pictograms
c) Bar diagrams
d) Line
diagrams
22. A pie diagram is also called:
a) Pictogram
b) Angular diagram
c) Line
diagram
d) Bar
diagram
23. In
volume diagram the three dimensions which are taken into account are:
a) Length,
weight, breadth
b) Height,
weight, breadth
c) Length, height, breadth
d) Length,
weight, height
24. The
median of a frequency distribution is found graphically with the help of:
a) Histogram
b) Frequency
curve
c) Frequency
polygon
d) Ogive
25. The mode of a frequency distribution can be determined
graphically by:
a) Histogram
b) Frequency
curve
c) Frequency
polygon
d) Ogive
26. Find the median of the given ogive:
a) 150
b) 200
c) 148
d) 175
27. What is the appropriate simple annual growth rate of
total assets between 1990 and 1993?
a) 36%
b) 12%
c) 9%
d) 27%
28. From the figure given in Question 27, find the only
item that has shown positive growth between 1991 and 1993?
a) Net
fixed assets
b) Net
current assets
c) Investments
d) Total
assets
29. If a sample of size n from a given finite population of
size N, then the total number of samples is:
a) N! /
(N –n)!
b) N!
c) N!
/n!
d) N! /n! (N – n )!
30. The set of
values of the statistic so obtained, one for each sample, constitutes what is
called: a) Sampling distribution
b) Systematic
sampling
c) Stratified
sampling
d) Cluster
sampling
31. Standard error of the sampling distribution of a
statistic t is:
a) √
Standard deviation
b) √Median
c) √Variance
d) √Mean
32. Convert the following into an ordinary frequency
distribution:
5
students get less than3 marks; 12 students get
less than 6 marks; 25 students get less than 9 marks; 33 students get less than
12 marks.
a) 0—3 3—6 6— 9
9 —12
5
7 13 8
b) 0—3 3—6
6— 9 9 —12
6
6
14 7
c) 0—3 3—6
6—9 9 —12
4 8 12 9
d) 0—3 3—6
6—9 9 – 12
6
8
12 7
33. From the below
given graph, find what expenditure for the 7 years together from percent of the
revenues together:
|
|
1989 1990 1991 1992 1993 1994 1995
a) 75%
b) 67%
c) 62%
d) 83%
34. From the above graph in which year was the growth in
expenditure maximum as compared to the previous year:
a) 1993
b) 1995
c) 1991
d) 1992
35. The equity base of the companies remains unchanged,
then the total divided earning by share holders in 1991-1992 is:
a) Rs104 lakh
b) Rs
9 lakh
c) Rs12.8 lakh
d) Rs15.6
lakh
36. From the above figure answer the retained profit in
1991-1992 as compared to that in 19901991 was:
a) Higher
by 2.5%
b) Higher
by 1.5%
c) Lower
by 2.5%
d) Lower by 1.5%
37. A professor keeps data on students tabulated by
performance and sex of the students. The data is kept on the computer disk and
due to virus the following data could be recovered. An expert committee was
formed and it was decided. Half the students were either excellent or good. 40%
of the students were female.1/3 of the male students were average.
|
Performance |
Total |
|||||
|
Average |
Good |
Excellent |
|
|
||
Male |
16 |
22 |
10 |
48 |
|
||
Female |
24 |
8 |
- |
32 |
|
||
Total |
40 |
30 |
10 |
80 |
|
||
How many students are both female and excellent:
a)
0
b)
8
c)
16
d)
32
38. Among
every student what is the ratio of male and female: a)
1:2
b) 2:1
c) 3:2
d) 2:3
39. Machine
A as well as machine B can independently produce either product P or Q. The
time taken by machine A and B in minutes to produce one unit of product P and Q
is given as follows: (each machine works 8 hours per day)
Product |
A |
B |
P |
10 |
8 |
Q |
6 |
6 |
If equal quantities of both are to be produced then out of
the 4 choices the least efficient way would be
a)
48 of each with 3 min idle
b)
64 of each with 12 min idle
c)
53 of each with
10 min idle
d)
71 of each with 9 min idle
40. If the number of units of P is to be 3 times that of Q,
what is the maximum idle time to maximize total
units manufactured?
a) 0 min
b) 24
min
c) 1
hr
d) 2
hr
UNIT 3 MEASURES OF STATISTICAL DATA
1. The standard
deviation for 15, 22, 27, 11, 9, 21, 14, 9 is:
a) 6.22
b) 6.12
c) 6.04
d) 6.32
2.
A student obtained the mean and the standard
deviation of 100 observations as 40 and 5.1. It was later found that one
observation was wrongly copied as 50, the correct figure being 40. Find the
correct mean and the S.D.
a) Mean
= 38.8, S.D =5
b) Mean = 39.9, S.D =5
c) Mean
= 39.9, S.D = 4
d) None
3.
The mean deviation about median from the data:
340, 150, 210, 240, 300, 310, 320 is: a) 51.6
b) 51.8
c) 52
d) 52.8
4. For a
frequency distribution mean deviation from mean is computed by a) ∑E
f /∑
E f |d|
b) ∑E
d /∑Ef
c) ∑E
fd/ ∑E
f
d) ∑Ef | d | / ∑E f
5. The mean deviation from the median is:
a) Equal
to that measured from another value
b) Maximum
if all the observations are positive
c) Greater
than that measured from any other value
d) Less than that measured from any value
6. The mean deviation of the series
a, a + d, a +2d……., a + 2n from its mean is a) (n + 1) d /2n +1
b) nd
/2n +1
c) n (n +1) d /2n +1
d) (2n
+1) d /n (n+1)
7.
A batsman score runs in 10 innings as 38, 70,
48, 34, 42, 55, 63, 46, 54 and 44. The mean deviation about mean is
a) 8.6
b) 6.4
c) 10.6
d) 7.6
8.
The arithmetic mean height of 50 students of a
college is 5’---8’. The height of 30 of
these is given in the frequency distribution. Find the arithmetic mean height
of the remaining 20 students.
Height in
inches: 5’---- 4” 5’--- 6” 5’ ---- 8” 5’----10” 6’ --- 0” Frequency: 4 12 4 8 2 a) 5’ ----8.8”
b) 5’
---- 8.0”
c) 5’-----
7.8”
d) 5’-----
7.0”
9. Find the sum of the deviation of
the variable values 3, 4, 6, 8, 14 from their mean a) 5
b) 0
c) 1
d) 7
10. The median of the observation 11, 12, 14, 18, x + 4,
30, 32, 35, 41 arranged in ascending order is 24, then x is
a) 21
b) 22
c) 23
d) 24
11. The median of the data: 19, 25, 59, 48, 35, 31, 30,
32, 51. If 25 is replaced by 52, what will be the new median.
a) 35
b) 53
c) 43
d) 45
12. If
the median of the following frequency distribution is 46, find the missing
frequencies.
Variable:
10—20 20—30 30—40
40---50 50—60 60---70
70---80 Total
Frequency: 12 30 a 65 b 25 18 229 a) a = 32
b =40
b) a
=31 b = 45
c) a
= 33 b = 42
d) a =34 b
=45
13. Find
the value of x, if the mode of the data is 25:
15, 20, 25, 18, 14, 15, 25, 15, 18, 16, 20,
25, 20, x,
a) 15
b) 18
c) 25
d) 20
14. Compute
the modal value for x : 95
105 115 125
135 145 155
165 175 f :
4 2 18
22 21 19
10 3 2
a) 175
b) 125
c) 145
d) 165
15. Compute the mode for the following frequency
distribution:
Size of items:
0-4 4-8 8-12
12-16 16-20 20-24
24-28 28-32 32-36
36-40
Frequency: 5
7 9 17 12
10 6 3 1 0 a) 32.66
b) 28.43
c) 24.87
d) 31.65
16. For the following grouped frequency distribution find
the mode:
Class: 3-6
6-9 9-12 12-15
15-18 18-21 21-24 Frequency: 2
5 10 23 21 12 3
a) 13.9
b) 14.7
c) 15.1
d) 14.6
17. The table shows the age distribution of cases of a
certain disease admitted during a year in a particular hospital.
Age (in years):
5-14 15-24 25-34
35-44 45-54 55-64
No of cases:
6 11 21 23 14
5
The average age for which maximum cases occurred is:
a) 34.33
b) 35.34
c) 36.31
d) 37.31
18. In a
moderately symmetric distribution mean, median and mode are connected by:
a) Mode
= 2 median – 3 mean
b) Mode
= 3 median – 4 mean
c) Mode = 3 median – 2 mean
d) Mode
= 2 median – 4 mean
19. The
mean of n observations is X. If k is added to each observation then the new
mean is
a) X
b) X + k
c) X –k
d) kX
20. The mean of n observations is X. If each observation
is multiplied by k, the mean of new observation is:
a) kX
b) X /k
c) X +k
d) X –
k
21. The
algebraic sum of the deviations of a set of n values from their mean is a) 0
b) n
– 1
c) n
d) n
+ 1
22. A,B,
C are three sets of values of x:
A: 2, 3, 7, 1, 3, 2, 3
B: 7, 5, 9, 12, 5,
3, 8
C: 4, 4, 11, 7, 2,
3, 4
Which is true:
a) Mean
of A = Mode of C
b) Mean
of C = Median of B
c) Median
of B = Mode of A
d) Mean, median, mode of A are equal
23. The mean and variance of 7 observations are 8 and 16 .
If 5 of the observations are 2, 4, 10, 12, 14 the remaining 2 observations are:
a) x =6 , y = 8
b) x=5,
y=7
c) x=7
, y=3
d) None
of these
24. The variance of 15 observations is 4. If each
observation is increased by 9, the variance of the resulting observation
is:
a) 2
b) 3
c) 4
d) 5
25. The mean of 5 observations is 4.4 and their variance
is 8.24. If 3 of the observations are 1, 2,
6. The other 2 observations are:
a) 9, 4
b) 7, 8
c) 6, 5
d) 4, 8
26. The geometric mean of 10 observation s on a certain
variable was calculated as 16.2. It was later discovered that one of the
observations was wrongly recorded as 12.9; in fact it was 21.9. The correct G.M
is:
a) 17.12
b) 18.43
c) 17.08
d) 18.15
27. Three groups of observations contain 8, 7 and 5
observations. Their geometric means are 8.52, 10.12 and 7.75. Find the
geometric mean of the 20 observations in the single group formed by pooling the
three groups is:
a) 7.831
b) 8.837
c) 9.643
d) 6.438
28. Find the Quartile deviation for the distribution:
Class Interval: 0
– 15 15 -30 30 – 45
45 – 60 60 – 75 75 – 90 90 – 105
f: 8 26 30 45 20 17 4 a) 15.44
b) 16.22
c) 14.55
d) 17.33
29. Find the quartile deviation for the data:
Income (in Rs.): Less than 50 50 -70
70 -90 90 – 110 110 -130
130 – 150 Above150 No of
Persons: 54 100 140
300 230 125 51 a) 18.625
b) 19.925
c) 17.485
d) None
of these
30. From the monthly income of 10 families find the
coefficient of range is:
S. No: 1 2 3 4 5 6 7 8 9 10 Income in (Rs.): 145
367 268 73
185 619 280
115 870 315 a) 0.1
b) 0.6
c) 0.84
d) 0.56
31. Find the value of third quartile if the values of
first quartile and quartile deviation
are 104 and 108 respectively.
a) 130
b) 140
c) 120
d) 110
32. Age distribution of 200 employees of a firm is given
below and calculate semi inter quartile range = (Q3 – Q1 ) /2 of the distribution:
Age in Years (less than): 25 30 35 40 45 50 55 No of Employees: 10 25 75 130 170 189 200 a) 4.75 years
b) 4.25
years
c) 4
years
d) None
of these
33. Find the lower quartile for the distribution
Wages: 0 – 10 10 – 20 20 – 30 30 -
40 40 – 50 No of Workers: 22 38 46 35 20 a) 13.80
b) 12.56
c) 14.803
d) None
of the above
34. Find the Mean deviation from the Mean for the
following
Class Interval: 0 – 10
10 – 20 20 – 30 30 – 40
40 -50 50 – 60 60 – 70 Frequency: 8 12 10 8 3 2 7 a) 14
b) 12
c) 15
d) 16
35. Mean deviation which is calculated is minimum at:
a) Mean
b) Median
c) Mode
d) All
the three
36. Initially there were 9 workers, all being paid a
uniform wage. Later a 10th worker is added to the list whose wage rate is Rs.
20 less than for others. The standard
deviations of wages for the group of 10 workers are:
a) 5
b) 4
c) 7
d) 6
37. Twenty passengers were found
ticketless on a bus. The sum of squares and the standard deviation of the
amount found in their pockets were Rs.2,000 and Rs.6. If the total fine imposed
on these passengers is equal to the total amount recovered from them and fine
imposed is uniform, what is the amount each one has to pay as fine? a) 5
b) 6
c) 8
d) 9
38. For any discrete distribution standard deviation is
not less than
a) Mean deviation from mean
b) Mean
deviation from median
c) Mode
d) None
of these
39. Mean
of 10 items is 50 and S.D is 14. Find the sum of squares of all items
a) 26543
b) 26960
c) 27814
d) 27453
40. Find
the range for the following data
14, 16, 16, 14, 22, 13, 15, 24, 12, 23, 14, 20, 17, 21,
22, 18, 18, 19, 20, 17, 16, 15, 11, 12, 21, 20, 17, 18, 19, 23.
a) 13
b) 12
c) 14
d) 16
UNIT 4 PERMUTATIONS, COMBINATIONS AND PROBABILITY
1. A five digit number is formed using
digits 1,3 5, 7 and 9without repeating any one of them. What is the sum of all
such possible numbers?
a) 6666600
b) 6666660
c) 6666666
d) None
of these
2. 139 persons have signed for an
elimination tournament. All players are to be paired up for the first round,
but because 139 is an odd number one player gets a bye, which promotes him to
the second round, without actually playing in the first round. The pairing
continues on the next round, with a bye to any player left over. If the
schedule is planned so that a minimum number of matches is required to
determine the champion, the number of matches which must be played is
a) 136
b) 137
c) 138
d) 139
3. A box contains 6 red balls, 7 green balls and 5 blue balls. Each
ball is of different size.
The probability that the red
ball selected is the smallest red ball is a) 1/8
b) 1/3
c) 1/6
d) 2/3
4. Boxes numbered 1,2,3,4 and 5 are kept in a
row, and they which are to be filled with either a red ball or a blue ball,
such that no two adjacent boxes can be filled with blue balls. Then how many
different arrangements are possible, given that all balls of given colour are
exactly identical in all respect?
a) 8
b) 10
c) 154
d) 22
5. For a scholarship, at the most n candidates out of 2n +
1 can be selected. If the number of different ways of selection of at least one
candidate is 63, the maximum number of candidates that can be selected for the
scholarship is
a) 3
b) 4
c) 6
d) 5
6. Ten
points are marked on a straight line and 11 points are marked on another
straight line. How many triangles can be constructed with vertices from among
the above points? a) 495
b) 550
c) 1045
d) 2475
7. There
are three cities A, B and C. Each of these cities is connected with the other
two cities by at least one direct road. If a traveler wants to go from one city
(origin) to another city (destination), she can do so either by traversing a
road connecting the two cities directly, or by traversing two roads, the first
connecting the origin to the third city and the second connecting the third
city to the destination. In all, there are 33routes from A to B (including
those via C), Similarly, there are 23 routes from B to C (including those via
A). How many roads are there from A to C directly? a) 6
b) 3
c) 5
d) 10
8. One
red flag, three white flags and two blue flags are arranged in line such that
i) No
two adjacent flags are of the same colour.
ii) The
flags at the two ends of the line are of different colours. In how many different ways the
flag be arranged?
a) 6
b) 4
c) 10
d) 2
9. Each
of the 11 letters A. H, I, M, O, T, U, V, W, X and Z appears same hen looked at
in the mirror. They are called symmetric letters. Other letters in the alphabet
are asymmetric letters. How many four letter computer passwords can be formed
using only the symmetric letters ( no repetition allowed)
a) 7920
b) 330
c) 146.40
d) 419430
10. An intelligence agency forms a code of two distinct digits
selected from 0, 1, 2,……, 9 such that the first digit of the code is non zero.
The code, handwritten on the slip, can create confusion, when read upside down
for example the code 91 can be read as 16. How many codes are there for which
no such confusion can arise? a) 80
b) 78
c) 71
d) 69
11. The
set of all possible outcomes of a random experiment is known as
a) Permutation
b) Combination
c) Probability
d) Sample space
12. A
card is drawn from a well shuffled pack of playing cards. Find the probability
that it is either a diamond or a king
a) 4/26
b) 4/13
c) 17/52
d) 16/13
13. Let
A and B be the two possible outcomes of an experiment and suppose P(A) =
0.4 P(AUB) =0.7 and
P(B) =p. For what choice of p are
A and B mutually exclusive?
a) 0.5
b) 0.2
c) 0.3
d) 0.6
14. Probability
that a man will be alive 25 years hence is 0.3 and the probability that his
wife will be alive 25 years hence is 0.4. Find the probability that 25 years
hence only the man will be alive will be
a) 0.12
b) 0.18
c) 0.28
d) 0.42
15. A
box of nine golf gloves contains two left-handed and seven right handed gloves.
If three gloves are selected without replacement, what is the probability that
all of them are left handed?
a) 1
b) 0
c) 7/18
d) 49/81
16. A
lady declares that by taking a cup of tea, she can discriminate whether the
milk or tea infusion was added to the cup. It is proposed to test this
assertion by means of an experiment with 12 cups of tea, 6 made in one way and
6 in the other, and presenting them to the lady for judgement in a random
order. The probability that on the null hypothesis that the lady has no
discrimination power, she would judge correctly all the 12 cups, it being known
to her that 6 are of each kind would be
a) 924
b) 1/925
c) 1/924
d) 925
17. A
restaurant serves two special dishes A and B to its customers consisting of 60%
men and 40% women. 80% of men order dish A and the rest B. 70% of women order B
and the rest A. In what ratio of A to B should the restaurant prepare the two
dishes?
a) 3:2
b) 2:3
c) 1:2
d) 2:1
18. A
card is drawn at random from a well shuffled pack of cards. The probability
that it is heart or a queen is
a) 4/13
b) 11/52
c) ½
d) 1/52
19. A
piece of electronic equipment has two essential parts A and B. In the past,
part A failed 30% of the times, part B failed 20% of the times and both failed
simultaneously 5% of the times. Assuming that both parts must operate to enable
the equipment to function, the probability that the equipment will function
is
a) 0.1
b) 0.52
c) 0.55
d) 0.15
20. In a
certain college, the students engage in sports in the following proportion
Football (F): 60% of all students Basketball (B): 50% of all students. Both
football and basketball: 30% of all students. If a student is selected at
random the probability that he will play neither sports is
a) 0.8
b) 0.10
c) 0.7
d) 0.20
21. If P(A) =1/4, P(B) =2/5 and P(AUB) =1/2 find P(Ac U Bc ), where
A and B are two non mutually exclusive events connected with a random
experiment E and Ac is the
complement event of A.
a) 0.85
b) 0.58
c) 0.80
d) 0.50
22. The
result of an examination given to a class on three papers A, B and C are 40%
failed in paper A, 30% failed in B, 25% failed in paper C, 15% failed in paper
A and B both. 12% failed in B and C both, 10% failed in A and C both, 3% failed
in A, B and C. What is the probability of a randomly selected candidates
passing in all three papers?
a) 0.6
b) 0.39
c) 0.56
d) 0.42
23. The
figure below shows the network of cities A, B, C, D, E and F. The arrows show
the permissible direction of travel. What is the number of distinct paths from
A to F?
a) 9
b) 10
c) 11
d) None
of these
24. Suppose
it is 11 to 5 against a person who is now 38 years of age living till he is 73
and 5 to
3against B, now 43 living till he is 78. The chance that
at least one of these persons will be alive
35 years hence is
a) 0.47
b) 0.57
c) 0.37
d) 0.67
25. The
problem in Mathematics is given to three students A, B and C whose chances of
solving it are 1/3, 1/4 and 1/2. The probability that the problem will be
solved is
a) 1/12
b) 3/4
c) 7/12
d) None
26. If
P(A) = 0.3 P(B) = 0.2
and P(C) =0.1 and A, B, C are
independent events the probability of occurrence of at least one of the
three events A,B, C is
a) 0.41
b) 0.37
c) 0.496
d) 0.387
27. A
speaks the truth 3 times out of 4, and B
7 times out of 10; They both assert that
a white ball has been drawn from a bag containing 6 balls of different colour.
The truth in the assertion is a) 35/36
b) 36/43
c) 25/36
d) 63/43
28. Three
urns are given, each containing red and white balls. Urn 1: 6 red balls and 4
white, Urn 2: 2 red and 6 white, Urn3: 1 red and 8 white. An urn is chosen at
random and a ball is drawn from this urn. The ball is red. The probability that
the urn chosen was urn 1 is
a) 196/173
b) 173/196
c) 173/198
d) 198/173
29. A
doctor is to visit a patient. The probability that he will come by car taxi
scooter or by other means of transport are 0.3, 0.2, 0.1 and 0.4. The
probabilities that he will be late are 1/4, 1/3 and ½, if he comes by car taxi
and scooter. But if he comes by other means of transport he will not be late.
When he arrives he is late. Therefore the probability that he comes by car are a) 1/2
b) 0
c) 1/4
d) 1
30. What
is the chance that a leap year selected at random will contain 53 Sundays? a) 2/7
b) 3/7
c) 1/7
d) 5/7
31. Out
of all the 2-digit integers between 1 and 200, a 2- digit number has to be
selected at random. What is the probability that the selected number is not
divisible by 7?
a) 11/90
b) 33/90
c) 55/90
d) 77/90
32. Amarnath
appears in an exam that has 4 subjects. The chance he passes an individual
subject’s test is 0.8. The probability that he will pass in at least one of the
subjects is a) 0.99984
b) 0.9984
c) 0.0004
d) None
of these
33. A
box contains 2 tennis , 3 cricket and 4 squash balls. Three balls are drawn in
succession with replacement. What is the probability that all are cricket
balls:
a) 1/27
b) 2/27
c) 3/27
d) 1/9
34. In a
garden 40% of the flowers are roses and the rest are carnations. If 25% of the
roses and 10% of the carnations are red the probability that a red flower
selected at random is a rose is a) 5/8
b) 2/8
c) 7/8
d) 3/8
35. Three
of the 6 vertices if a regular hexagon are chosen at random. The probability
that the triangle with these vertices is equilateral is
a) 1/10
b) 4/10
c) 3/10
d) 1/5
36. What
is the value of n(P(P(P(ø))))
a) 3
elements
b) 4 elements
c) 8
elements
d) 5
elements
37. In
how many ways can 10 identical presents be distributed among 6 children so that
each child gets at least one present ?
a) 15
C6
b) 16
C6
c) 9 C5
d) 610
38. There
are 6 pups and 4 cats. In how many can they be seated in a row so that no cats
sit together:
a) 6
! 6 X 6 ! 6
b) 10!/4!6!
c) 6! X 7P4
d) 6!7!
39. There
are V lines parallel to the X axis and W lines parallel to the Y axis. How many
rectangles can be formed with the intersection of these lines?
a) vP2
.w P2
b) vC2 . w C2
c) vwC2
d) None
of these
40. From
4 men and 4 women a committee of 5 is to be formed. Find the number of ways of
doing so if the committee consists of a president, a vice president and three
secretaries? a) 720
b) 450
c) 1120
d) None
of these
UNIT 5 RANDOM VARIABLES AND DISTRIBUTION FUNCTIONS
1. If the
probability density of X is given by f(x) =
2xe-x² for x>0
0 elsewhere
and
Y = X2
The probability density of Y is
a) g(y) = e-y for
y > 0 and g(y) elsewhere
b) g(y)
= ey for y > 0 and g(y) = 0
c) g(y)
= e-y for y< 0 and g(y) > 0
d) None
of these
2. If
X has the uniform density with the parameters α = 0 and β = 1. Find the
probability density of the random variable Y = √X
a) g(y)
= y for 0 < y < 1 and g(y) = 0 elsewhere
b) g(y) = 2y for 0 < y < 1 and g(y) = 0 elsewhere
c) g(y)
= 2y for 0 > y > 1 and g(y) = 0 elsewhere
d) None
of these
3. If
X1 and X2 are independent random variables having exponential densities with
the parameters a and b the probability density of Y = X1+ X2 when a ≠
b
a) f(y)
= 1/a+b. (e-y/a – e-y/b ) for y > 0 and f(y) = 0 elsewhere
b) f(y)
= 1/a-b. (e-y/a – e-y/b ) for y < 0 and f(y) = 1 elsewhere
c) f(y) = 1/a-b. (e-y/a – e-y/b ) for y > 0 and f(y)
= 0 elsewhere
d) None
of these
4. If
X is the number of head obtained in 4 tosses of a balanced coin then find the
probability distribution of the random variable Z = (X-2)2
a) z
0 1 4 h(z) 3/8
4/8 1/8
b) z 0
1 4 h(z)
1/8 4/8 1/8
c) z 0
1 4 h(z)
3/8 2/8 1/8
d) z 0
1 4 h(z)
3/8 7/8 1/8
5. If the
joint density of X1 and X2 is given by
f(x, x2 ) = 6e-3x1 -2x2 for x1 > 0
x2 > 0
0 elsewhere
Find the probability density of Y = X1+ X2
a) f(y)
= 6(ey – e-3y ) for y < 0 elsewhere
f(y) = 0
b) f(y) = 6(e-2y
– e-3y ) for y > 0 elsewhere f(y) = 0
c) f(y)
= 6(e-2y – e-y ) for y > 0 elsewhere
f(y) = 1
d) f(y)
= 6(e-2y – e-y/2 ) for y > 0
elsewhere f(y) = 0
6. If
X has a hypergeometric distribution with M = 3, N = 6 and n = 2, find the
probability distribution of Y, the number of successes minus the number of
failures
a) h(0) = 1/5 , h(1) = 3/5 , h(2) = 1/5
b) h(0)
= 2/5 , h(1) = 3/8 , h(2) = 1/5
c) h(0)
= 9/5 , h(1) = 3/5 , h(2) = 1/5
d) h(0)
= 1/5 , h(1) = 4/5 , h(2) = 1/5
7. If the
probability density is given by
f(x) = kx3 /(1 + 2x)6 for x> 0
0 elsewhere
Where k is appropriate constant the probability density of
the random variable Y = 2X / 1 +
2X
a) g(y)
= k/16y3 .(1-y) for 0 > y > 1 and g(y) = 0 elsewhere
b) g(y) = k/16y3 .(1-y) for 0 < y < 1 and g(y) = 0
elsewhere
c) g(y)
= k/16y2 .(1-y) for 0 < y < 1 and g(y) = 0 elsewhere
d) g(y)
= k/16y9 .(1-y) for 0 < y < 1 and g(y) = 1 elsewhere
8. Two
dices are thrown simultaneously and ‘getting a number less than 3’ on a die is
termed as a success. Obtain the probability distribution of the number of
successes
a) x
0 1 2 p(x) 4/9 5/9 1/9 b) x
0 1 2 p(x) 1/9 4/9 1/9 c) x
0 1 2 p(x) 4/9 4/9 1/9 d) x
0 1 2 p(x) 4/9
7/9 1/9
9. Obtain
the probability distribution of the number of sixes in 2 tosses of dice a)
x 0 1 2
p(x) 4/9 4/9 1/9
b) x 0 1 2 p(x) 4/72
1/9 1/9
c) x 0 1 2 p(x) 4/9 4/36 8/9
d) x
0 1 2
p(x) 25/36 10/36 1/36
10. Three
cards are drawn at random successively, with replacement, from a well shuffled
pack of cards. Getting a card of ‘diamonds’ is termed as success. Obtain the
probability distribution of the number of successes.
a) x
0 1 2 3 p(x)
27/64 27/64 9/64 1/64
b) x 0 1 2 3 p(x) 1/9 4/9 1/9 6/9
c) x 0 1 2 3 p(x) 4/9 4/9 1/9 5/9
d) x 0 1 2 3 p(x) 4/64 7/64 1/64 8/64
11. A
die is thrown at random. What is the expectation of the number on it:
a) 3.7
b) 3.1
c) 3.5
d) 3.8
12. What
is the expected number of heads appearing when a fair coin is tossed three
times? a) 2.1
b) 1.5
c) 3.2
d) 4.1
13. A
contractor spends Rs. 3,000 to prepare for a bid on a construction project
which, after deducting manufacturing expenses and the cost of bidding, will
yield a profit of Rs. 25,000 if the bid is not won. If the chance of winning
the bid is 10%, compute his expected profit?
a) 100
b) 607
c) 35
d) 200
14. Determine
which of the following given values can serve as the values of a probability
distribution of a random variable with the range x = 1, 2, 3 and 4
a) f(1)
= 0.25 , f(2) = 0.75 , f(3) = 0.25 , f(4) = -0.25
b) f(1) = 0.15 , f(2) = 0.27 , f(3) = 0.29 , f(4) = 0.29
c) f(1)
= 1/19 , f(2) = 10/19 , f(3) = 2/19 , f(4) = 5/19
d) None
of these
15. For
what values of k can f(x) = (1-k) kx
a) 0<k<1
b) k=0
c) k>1
d) None
of these
16. From
a bag containing 4 white and 6 red balls, three balls are drawn at random and
if each white ball drawn carries a reward of Rs4 and each red ball Rs6, find
the expected reward of the draw
a) Rs14.8
b) Rs15.6
c) Rs31
d) Rs16
17. A
lot of 12 television sets include 2 with white chords. If 3 of the sets are
chosen at random for shipment to the hotel, how many sets with white chords can
the shipper expect to send to the hotel
a) 0
b) 1
c) 1/2
d) All
of the above
18. The
joint probability density function
f(x,y) = 3/5x(y+x) for 0<x<1 0<y<2
0 elsewhere
Of 2 random variables X and Y, find
P[(X,Y)€A] where A is the region (x,y)/0 < x, ½, 1<y<2 a) 11/65
b) 11/80
c) 10/76
d) 67/80
19. E(x2)
= 91/6. Find the value of E(2 x2+1) is
a) 92/3
b) 91/3
c) 90/3
d) 94/3
20. If the
probability density of X is given by f(x) =
2(1-x) for 0<x<1
0 elsewhere
To
evaluate E[(2X+1)2]
a) 2
b) 1
c) 4
d) 3
21. If X
has the probability density
f(x) = ex for x>0
0 elsewhere
Find
the expected value of g(X) = e3x/4
a) 1
b) 2
c) 3
d) 4
22. Given
that X has the probability distribution f(x) = 1/8(3/x) for x = 0, 1, 2 and 3,
find the moment-generating function of this random variable and use it to
determine µ1`and µ2 ` a) 0
b) 3/2
c) 1/2
d) 1
23. For
any random variable for which E(x) exists find the value of µ0 a) 0
b) -1
c) 2
d) 1
24. Find
variance for the random variable x that has the probability density
f(x)
= x/2 for 0<x<1
0 elsewhere
a) 1/9
b) 2/9
c) 4/9
d) 5/9
25. Find
µ1`of the discrete random variable x that has the probability distribution f(x)
= 2(1/3x) for x = 1, 2, 3- - -
a) 1/2
b) 0
c) 1
d) 3/2
26. The
moment-generating function of a random variable which has probability density
f(x) =
1/2e-|x| for - ∞ < x < ∞ is
a) Mx
(t) = 1/2t+1
b) Mx (t) = 1/1-t2
c) Mx
(t) = 1/-2t
d) Mx
(t) = 1/t2
27. Find
the E(X) whose probability density is given by f(x) = 1/8(x+1) for 2<x<4
0 elsewhere
a) 35/12
b) 38/12
c) 37/12
d) 33/12
28. If the
joint probability density of X and Y is given by f(x,y) = 2/7(2y+x) for 0<x<1 1<y<2
0 elsewhere
Find the
expected value of g(X,Y) = X/Y3
a) 13/84
b) 15/84
c) 84/13
d) 84/15
29. If
the probability density of Xs given by
x/2 for 0<x≤1 f(x) = 1/2
for 1<x≤2 3-x/2 for 2<x<3
0 elsewhere
Find the
expected value of g(X) = X2-5X+3
a) |
11/3 |
b) |
-11/3 |
c) |
-11/6 |
d) |
11/6 |
30. Suppose
an insurance company offers a 45 year old man a Rs1,000. 1 year term insurance
policy for an annual premium of Rs12 . Assuming that the number of deaths per
1000 is 5 for persons in this age this group. The expected gain for the
insurance company on a policy of this type is
a) 7
b) 8
c) 9
d) 10
31. In a
business venture a man can make a profit of Rs 2,000 with probability of 0.4 or
have a loss of Rs 1,000 with a probability of 0.6. His expected profit is a) 100
b) 200
c) 400
d) 300
32. In a
random throw of n dice, the expectation of the sum of points on them is a) n/2
b) 3n/2
c) 7n/2
d) 9n/2
33. A
number is chosen at random from the set 10.11,12- - -109; and another number is
chosen at random from the set 12,13 ,14- - - 61. The expected value of their
sum is a) 95
b) 96
c) 97
d) 98
34. Three
coins whose faces are marked 1 and 2 are tossed. Their expectations of the
total values of numbers on their faces is
a) 9.5
b) 4.5
c) 3
d) 4
35. If X
has the probability density
f(x) = k.e-3x for x>0
0 elsewhere Find k and P(0.5≤
X ≤
1)
a) 0.173
b) 0.5
c) 0.11
d) None
of these
36. A
and B throw with one die for a prize of Rs199 which is to be won by the player
who first throws 6. If A has the first throw their respective expectation
are
a) Rs
64, Rs 46
b) Rs 54, Rs 45
c) Rs
87, Rs 78
d) Rs
35, Rs 53
37. When
2 unbiased coins are tossed once, the variance of the number of head is a) 1
b) 3/2
c) 1/4
d) None of these
38. A
dice is tossed twice ‘getting a number less than 3’ is termed as success. Hence
the mean of the number of successes is
a) 1
b) 3/2
c) 1/4
d) 2/3
39. The
expected value of X is usually
written as:
a) E(X)
or Σ
b) E(X) or µ
c) E(X)
or Ï•
d) E(X)
or λ
40. The
probability distribution for
x :
8 12 16 20 24
p(x) :
1/8 1/6 3/8 ¼ 1/12
The
variance of the random variable x is
a) 20
b) 21
c) 22
d) 23
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