# Probability & Statistics MCQs Part I

1) In an entrance test there are multiple choice questions.
There are four possible answers to each question of which one is correct.
The probability that a student knows the answer to a question is 90%. If he
gets correct answer to a question, then the probability that he was guessing
is

a)
b)
c)
d)

2) A letter is taken out at random from “ASSISTANT”
and another taken out from “STATISTICS”. The probability that they
are the same letters is

a)
b)
c)
d) none of these

3) A bag contains a white and b black balls. Two players
A and B alternately draw a hall from the bag, replacing the ball each time after
the draw. A begins the game. If the probability of A winning (that is drawing
a white ball) is twice the probability of B winning, then the ratio a:b is equal
to

a) 1 : 2
b) 2 : 1
c) 1 : 1
d) none of these

4) A sum of money is rounded off to the nearest rupee.
The probability that rounded off error is at least ten paise is

a)
b)
c)
d)

5) Two persons each makes a single throw with a pair
of dice. The probability that the throws one unequal is given by

a)
b)
c)
d) none of these

6) A bakery makes 80 loaves of bread daily. Ten of them
are underweight. An inspector weighs 5 loaves at random. The probability than
an underweight loaf will be discovered is

a) 0.4965
b) 0.8532
c) 0.3024
d) none of these

7) In a family of n children, let A be the event that
the family has children of both sexes and let B be the event that there is at
most one girl in the family. Then the value of n for which the event A and B
are independent is (assuming that each child has probability
of being a boy)

a) 3
b) 4
c) 5
d) 6

8) Two absent minded room mates, mathematicisns, forget
their umbrellas in some way or another. A always takes his umbrella when he
goes out, while B forget to take his umbrella with the probability .
Each of them forgets his umbrella at shop with probability .
After visiting three shops they return home. The probability that they have
only one umbrella is:

a)
b)
c)
d) none of these

9) A parent particle can be divided into 0, 1 or 2 particles
with probabilities . It disappears
after splittting. Beginning with one particle, the progenitor, let us denote
by the number of particles in
the i generation, then is equal
to

a)
b)
c)
d) none of these

10) Two gamblers. A and B agree to play as follows. They
throw two dice and if the sum S of the outcomes is < 10, B receives S rupees from A, otherwise B pay to A x rupees, then the value of x so that the game is fair is:

a)
b) Rs.100
c)
d) none of these

11) The mean monthly salary of the employees in a certain factory is Rs.500. The mean monthly salary of male and female employees are respectively Rs.510 and Rs.460. The percentage of male employees in the factory is
a) 60
b) 70
c) 80
d) 90

12) If the mean of the set of numbers is
then
the mean of the numbers
is

a)
b)
c)
d) none of these

13) The weighted mean of first n natural numbers whose weights are equal to the
squares of corresponding numbers is

a)
b)
c)
d)

14) The arithmetic mean of 9 observations is 100 and that of 6 is 80, the combined
mean of all the 15 observations will be

a) 100
b) 80
c) 90
d) 92

15) If the difference between mean and mode is 63 the difference between mean
and median is

a) 189
b) 21
c) 31.5
d) 48.5

16) If a variable takes the discrete values
then the median is

a)
b)
c)
d)

17) If the mean of the distribution
is 2.6, then the value of k is

a) 8
b) 10
c) 12
d) 18

18) The standard deviation of first n natural numbers is
a)
b)
c)
d) None of these

19) Mean deviation of the series
a)
b)
c)
d)

20) A sample of 35 observations has the mean 80 and S.D. as 4. A second sample
of 65 observations from the same population has mean 70 and S.D.3. The S.D.
of the combined sample is

a) 5.85
b) 5.58
c) 34.2
d) None of these

21) The mean and S.D. of the marks of 200 candidates were found to be 40
and 15 respectively. Later, it was discovered that a score 40 was wrongly read
as 50. The correct mean and S.D. respectively are

a) 14.98, 39.95
b) 39.95, 14.98
c) 39.95, 224.5
d) None of these

22) The coefficient of correlation between two variables x and y is 0.5 their
covariance is 16 and S.D. of x is 4, then the S.D. of y is

a) 4
b) 8
c) 16
d) 64

23) For the observations {(1,2), (2, 5), (3, 7), (4, 8), (5, 10)} Karl-Pearsons
coefficient of correlation is given by

a) 0.75
b) 0.85
c) 0.52
d) 0.985

24) If then the cov. is equal to
a) 22
b) 2
c) – 2
d) none of these

25) If the sum of squares of rank-differences in Hindi and English marks of 10 students is 150, the coefficient of rank correlation is
a) 0.909
b) 0.091
c) 0.849
d) None of these

26) Let Then estimate of value of y corresponding to is
a) 110
b) 120
c) 100
d) None of these

27) Two random variables have the least square regression lines The coefficient of correlation between is given by
a) -0.5
b) 0.5
c) 0.25
d) None of these

28) Which of the following is not a measure of central tendency.
a) Mean
b) Median
c) Mode
d) Range

29) The mean of first n natural numbers is
a)
b) n(n+1)
c)
d) (n+1)

30) For a continuous series the mean is computed by the following formula
a)
b)
c)
d)

31) The mean of the squares of first n natural numbers is
a)
b)
c)
d)

32) If the mean of the set of numbers is, then the mean of the numbers is
a)
b)
c)
d)

33) If the mean of 3,4, x,7, 10 is 6, then the value of x is
a) 4
b) 5
c) 6
d) 7

34) The weighted mean of first n natural numbers whose weights are equal to the squares of corresponding numbers is
a)
b)
c)
d)

35) The mean of a set of numbers is . If each number is increased by , the mean of the new set is
a)
b)
c)
d) none of these

36) If the mode of a data is 18 and the mean is 24, then median is
a) 18
b) 24
c) 22
d) 21

37) The mean of a set of numbers is x. If each number is multiplies byλ, then the mean of the new set is
a)
b)
c)
d) none of these

38) The median of 10, 14, 11, 9,8,12,6 is
a) 10
b) 12
c) 14
d) 11

39) If mean= (3 median-mode) k, then the value of k is
a) 1
b) 2
c)
d)

40) If the mean of numbers 27,31,89,107,156 is 82, then the mean of 130, 126,68,50,1 is
a) 75
b) 157
c) 82
d) 80

41) The A.M. of a set of 50 numbers is 38. If two numbers of the set, namely 55 and 45 are discarded, the A.M. of the remaining set of numbers is
a) 38.5
b) 37.5
c) 36.5
d) 36

42) An automobile driver travels from plane to a hill station 120 km distant at an average speed of 30 km per hour. He then makes the return trip at an average speed of 25 km per hour. He covers another 120 km distance on plane at an average speed of 50 km per hour. His average speed over the entire distance of 300 km will be
a)
b)
c)
d) none of these

43) The central value of the set of observations is called
a) Mean
b) Median
c) Mode
d) G.M

44) The mean of first three terms is 14 and mean of next two terms is 18. The mean of all the five terms is
a) 14.5
b) 15.0
c) 15.2
d) 15.6

45) If the mean of 1,2,3……….n is
then n is

a) 10
b) 12
c) 11
d) 13

46) A group of 10 items has mean 6. If the mean of 4 of these items is 7.5, then the mean of the remaining items is
a) 6.5
b) 5.5
c) 4.5
d) 5.0

47) The mean of a set of observation of . If each observation is divided by a, and then is increased by 10, then the mean of the new set is
a)
b)
c)
d)

48) The number of observations in a group is 40. If the average of first 10 is 4.5 and that of the remaining 30 is 3.5, then the average of the whole group is
a)
b)
c) 4
d) 8

49) The mean of the values 0,1,2………..n having corresponding weight
respectively is

a)
b)
c)
d)

50) The reciprocal of the mean of the reciprocals of n observations is their
a) A.M
b) G.M
c) H.M
d) none of these

51) A car completes the first half of its journey with a velocity of v1 and the rest half with a velocity v2. Then the average velocity of the car for the whole journey is
a)
b)
c)
d) none of these

52)

 The mode of the distribution. Marks 4 5 6 7 8 No. of students 6 7 10 8 3 is:

a) 5
b) 6
c) 8
d) 10

53) A set of numbers consists of three 4`s , five 5`s, six 6`s,
eight 8`s and seven 10`s. The mode of this set of numbers is

a) 6
b) 7
c) 8
d) 10

54) The mean deviation of the numbers 3,4,5,6,7 is
a) 0
b) 1.2
c) 5
d) 25

55) The mean age of a combined group of men and women is 30 years. If the means of the age of men and women are respectively 32 and 27, then the percentage of women in the group is
a) 30
b) 40
c) 50
d) 60

56) If the algebraic sum of deviations of 20 observations from 30 is 20, then the mean of observations is
a) 30
b) 30.1
c) 29
d) 31

57) Mode is approximately given by
a) 2 median -3 mean
b) 2 median +3 mean
c) 3 median -2 mean
d) 3 median +2 mean

58) A student obtain 75%, 80% and 85% in three subjects. If the marks of another subject are added, then the average cannot be less than
a) 60%
b) 65%
c) 80%
d) 90%

59) Which one of the following is a source of data for primary investigations?
a) magazines
b) newspapers
c) government publications
d) questionnaires

60) Mean of 100 items is 49. It was discovered that three items which should have been 60, 70, 80 were wrongly read as 40, 20, 50 respectively. The correct mean is
a) 48
b) 82.5
c) 50
d) 80

61) The A.M. of the series, 1,2,4,8,16,……………. is
a)
b)
c)
d)

62) The A.M of n observations is M. If the sum of n-4 observations is a, then the mean of remaining 4 observations is
a)
b)
c)
d) nM+a

63) A person purchases one kg. of tomatoes from each of the 4 places at the rate of 1 Kg, 2 Kg, 3 Kg and 4 Kg per rupee respectively. On the average he has purchased x kg of tomatoes per rupee, then the value of x is
a) 2.5
b) 1.92
c) 2
d) none of these

64) The mean income of a group of workers is and that of another group is . If the number of workers in the second group is 10 times the number of workers in the first group, then the mean income of the combined group is
a)
b)
c)
d)

65) If X and Y are two variables such that
then

a) 1/2
b) -1/2
c) 1/4
d) -1/4

66) If a linear relation aX+bY+c=0 exists between the variables X and Y and
<0,
then the coefficient of correlation between X and Y is

a) 1
b) -1
c) 0
d) any number between -1 and 1

67) If a linear relation aX+bY+c=0 exists between the variables X and Y and
>0,
then the coefficient of correlation between X and Y is

a) 1
b) -1
c) 0
d) any number between -1 and 1

68) Let X and Y be two variables with the same variance and U and V be two variables
such that U = X +Y, V = X – Y Then Cov (U,V) is equal to

a) Cov ( X,Y)
b) 0
c) 1
d) -1

69) Let X and Y be two variables with the same variance and U and V be two variables
such that U = X +Y, V = X – Y. Then r(X,Y) is equal to

a) 1
b) -1
c) 0
d) none of these

70) If Z = aX + bY and r is the correlation coefficient between X and Y, then
is equal to

a)
b)
c)
d) none of these

71) The coefficient of correlation r between two variables X and Y is given by
a)
b)
c)
d) none of these

72) If X and Y are two uncorrelated variables and if u = X + Y,
then r (U,V) is equal to

a)
b)
c)
d) none of these

73) If X and Y are two independent variables with means 5 and 10 and variances
4 and 9 respectively. If U = 3X + 4Y and V = 3X – Y, then r (U,V)is equal to

a) 0
b) 1
c)
d) none of these

74) Let X and Y be two independent variables with variances 36 and 16 respectively,

then the coefficient of correlation between U = X + Y and V = X -Y is

a)
b)
c)
d) none of these

75) If X and Y are two variables such that
then

a)
b)
c)
d) none of these

76) The coefficient of correlation between X and Y is 0.60 Their covariance
is 4.8, Var (X) = 9. Then
is

a) 8/3
b) 3/8
c) 8/9
d) none of these

77) If the coefficient of rank correlation between marks in mathematics
and marks in physics obtained by a certain group of students is 0.8. If
the sum of the squares of the differences in ranks is given to be 33,
then the number of students in the group is

a) 11
b) 10
c) 30
d) none of these

78) Let
be the ranks of n individuals according to character A and
the ranks of the same individuals according to other character B such that

for
Then the coefficient of rank correlation between the characters A and B is

a) 1
b) 0
c) -1
d) none of these

79) If X and Y are independent variables, then the two lines of regression are
a) x = 0, y = 0
b) x = 0,y = const
c) x = const, y=0
d) x = const, y = const

80) If the lines of regression of Y on X and X on Y are respectively
and ,
Then

a)
b)
c)
d) none of these

81) If
are regression coefficients of Y on X and X on Y respectively, then

a)
b)
c)
d) none of these

82) If
are both positive, then

a)
b)
c)
d) none of these

83) If the lines of regression of Y on X and X on Y are respectively y = Kx
+ 4 and x = 4y + 5, then

a)
b)
c)
d) none of these

84) If the lines of regression in a bivariate distribution are given by
then the coefficient of correlation is

a) -0.866
b) 0.866
c) -0.666
d) 0.666

85) The line of regression of X on Y referred to the means of X and Y as the
origins is

a)
b)
c)
d) none of these

86) If the correlation coefficient between two variables X and Y is 0.4 and
the regression coefficient of X on Y is 0.2, then the regression coefficient
of Y on X is

a) 0.4
b)
c) 0.8
d) none of these

87) If the regression coefficient of Y on X is
, then the regression coefficient of X on Y

a)
b)
c) is less than 1
d) can take any value

88) The point that lies on both the lines of regression for a bivariate
distribution is

a) origin
b)
c)
d)

89) Angle between two lines of regression is given by
a)
b)
c)
d)

90) If
is the angle between the two regression lines with correlation coefficient
r, then

a)
b)
c)
d)

91) If the coefficient of correlation between X and Y is 0.28, covariance
between X and Y is 7.6 and the variance of X is 9, then the S.D. of Y
series is

a) 9.8
b) 10.1
c) 9.05
d) 10.05

92) Let X and Y be two variables with correlation coefficient r. If the
values of X and Y series are changed such that the Cov (X,Y) remains unchanged
while the variances of X and Y become 4 times their original values, then
the correlation coefficient between X and Y becomes

a) 4r
b)
c) 16r
d)

93) If the two lines of regression are x + 4y = 3 and 3x + y = 15, then
the value of x for y = 3 is

a) 4
b) -9
c) -4
d) none of these

94) Let X and Y be two variables with the same mean. If the lines of regressions
of Y on X and X on Y are respectively y = ax + b and
, Then the value of the common mean is

a)
b)
c)
d)

95) Let X and Y be two variables with the same mean. If the lines of regressions
of Y on X and X on Y are respectively y = ax + b and ,
then value of
is

a)
b)
c)
d)

96) Regression of savings (S) of a family on income Y may be expressed as

where a and m are constants. In a random sample of 100 families the variance
of savings is one-quarter of the variance of incomes and the correlation
coefficient is found to be 0.4. The value of m is

a) 2
b) 5
c) 8
d) none of these

97) The r(x, y) between x and y 0.5, cov(x, y)=20 and
a) 4
b) 8
c) 10
d) 64

98)
a) 22
b) 2
c) -2
d) none of these

99) The sum of squares of differences in ranks of marks obtained in Physics
and Chemistry of 10 students in a test is 25, then the coefficient of rank
correlation is

a) 0.15
b) 0.85
c) 0.5
d) 0.2

100) The regression coefficient of y on x is
. If the acute angle between the regression lines is

a)
b)
c)
d)