Class 11 Mathematics Statistics MCQs Set 06

Question. The mean marks got by 300 students in the subject of statistics was 45. The mean of the top 100 of them was found to be 70 and the mean of the last 100 was known to be 20, then the mean of the remaining 100 students is
(a) 45
(b) 58
(c) 68
(d) 88
Answer: (a) 45

Question. The average marks of boys in a class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is AIEEE - 2007
(a) 60
(b) 40
(c) 20
(d) 80
Answer: (d) 80

Question. Mean of ‘n’ items is \( \overline{x} \). If these n items are successively increased by \( 2, 2^2, 2^3, \dots, 2^n \), then the new mean is
(a) \( \overline{x} + \frac{2^{n+1}}{n} \)
(b) \( \overline{x} + \frac{2^{n+1} - 2}{n} \)
(c) \( \overline{x} + \frac{2^n}{n} \)
(d) \( \overline{x} + 2^n \)
Answer: (b) \( \overline{x} + \frac{2^{n+1} - 2}{n} \)

Question. The frequency distribution of discrete data given below, the frequency x against value 0 is missing.
Variable x : 0 1 2 3 4 5
Frequency f : x 20 40 40 20 4
If the mean is 2.5, then the missing frequency x will be____
(a) 0
(b) 1
(c) 3
(d) 4
Answer: (d) 4

Question. The minimum value of \( (x - 8)^2 + (x + 4)^2 + (x - 3)^2 + (x - 6)^2 + (x + 3)^2 \) is
(a) 114
(b) 141
(c) 104
(d) 2
Answer: (a) 114

Question. Product of n positive numbers is unity. The sum of these numbers cannot be less than
(a) 1
(b) n
(c) \( n^2 \)
(d) 2
Answer: (b) n

Question. An automobile driver travels from plane to hill station 100 km distance at an average speed of 30 km per hour. He then makes the return trip at average speed of 20 km per hour. What is his average speed over the entire distance (200 km)?
(a) 25 km/hr
(b) 24.6 km/hr
(c) 24 km/hr
(d) 24.5 km/hr
Answer: (c) 24 km/hr

Question. If A.M. = 24.5, G.M. = 24.375 then H.M. =
(a) 24
(b) 24.125
(c) 24.5
(d) 24.25
Answer: (d) 24.25

Question. The minimum value of \( | x - 6 | + | x + 3 | + | x - 8 | + | x + 4 | + | x - 3 | \) is
(a) 11
(b) 21
(c) 31
(d) 42
Answer: (b) 21
Solution:
Minimum value obtained at median of -4, -3, 3, 6, 8

Question. If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately
(a) 20.5
(b) 22.0
(c) 24.0
(d) 25.5
Answer: (c) 24.0

Question. Mean deviation of the series a, a+d, a + 2d,--------, a + 2nd from its mean is
(a) \( \frac{(n+1)d}{(2n+1)} \)
(b) \( \frac{nd}{2n+1} \)
(c) \( \frac{(2n+1)d}{n(n+1)} \)
(d) \( \frac{n(n+1)d}{2n+1} \)
Answer: (d) \( \frac{n(n+1)d}{2n+1} \)

Question. If mean deviation through median is 15 and median is 450, then coefficient of mean deviation is
(a) 1/30
(b) 30
(c) 15
(d) 45
Answer: (a) 1/30

Question. The mean and S.D. of 1, 2, 3, 4, 5, 6 is
(a) \( 3, 3 \)
(b) \( \frac{7}{2}, \sqrt{\frac{35}{12}} \)
(c) \( \frac{7}{2}, \sqrt{3} \)
(d) \( \frac{35}{12} \)
Answer: (b) \( \frac{7}{2}, \sqrt{\frac{35}{12}} \)

Question. If the S.D. of n observations \( x_1, x_2, \dots, x_n \) is 4 and another set of n observations \( y_1, y_2, \dots, y_n \) is 3 the S.D. of n observations \( x_1-y_1, x_2-y_2, \dots, x_n-y_n \) is
(a) 1
(b) \( 2 / \sqrt{3} \)
(c) 5
(d) 7
Answer: (c) 5

Question. The variance of first 10 multiples of 3 is
(a) 64.25
(b) 54.25
(c) 70.25
(d) 74.25
Answer: (d) 74.25

Question. Let r be the range and \( S^2 = \frac{1}{n-1}\sum (x_i - \overline{x})^2 \). If \( S^2 \le r^2 k \) then k is equal to
(a) \( \frac{1}{n-1} \)
(b) \( \frac{n}{n-1} \)
(c) \( \frac{n+1}{2(n-1)} \)
(d) \( \frac{1}{2(n-1)} \)
Answer: (b) \( \frac{n}{n-1} \)

Question. The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80, then which of the following gives possible values of a and b (AIEEE-2008)
(a) a = 0, b = 7
(b) a = 5, b = 2
(c) a = 1, b = 6
(d) a = 3, b = 4
Answer: (d) a = 3, b = 4

Question. Suppose a population A has 100 observations 101, 102....., 200 and another population B has 100 observations 151, 152, ..... 250. If \( V_A \) and \( V_B \) represent the variances of the two populations, respectively, respectively, then \( V_A / V_B \) is
(a) 1
(b) 9/4
(c) 4/9
(d) 2/3
Answer: (a) 1

Question. If the mean deviation about the median of the numbers a, 2a, ........, 50a is 50, then |a| equal to
(a) 4
(b) 5
(c) 2
(d) 3
Answer: (a) 4

Question. The variance of first 50 even natural numbers is
(a) 833 / 4
(b) 833
(c) 437
(d) 437 / 4
Answer: (b) 833

Question. All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given?
(a) mode
(b) variance
(c) mean
(d) median
Answer: (b) variance

Question. If a variable takes the values 0, 1, 2, ..... n with frequencies proportional to the binomial coefficients \( {}^nC_0, {}^nC_1, ......, {}^nC_n \) then the mean of the distribution is
(a) \( \frac{n(n+1)}{4} \)
(b) \( \frac{n}{2} \)
(c) \( \frac{n(n-1)}{2} \)
(d) \( \frac{n(n+1)}{2} \)
Answer: (b) \( \frac{n}{2} \)

Question. If a variable takes values 0, 1, 2, ....., n with frequencies proportional to \( q^n, {}^nC_1 pq^{n-1}, {}^nC_2 p^2 q^{n-2}, ......, p^n \) where p+q=1 then the mean is
(a) np
(b) nq
(c) npq
(d) \( np^2 \)
Answer: (a) np

Question. 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) \( \frac{nM+a}{4} \)
(b) \( \frac{nM-a}{4} \)
(c) \( \frac{nM-a}{2} \)
(d) n M + a
Answer: (b) \( \frac{nM-a}{4} \)

Question. A distribution consists of three components with frequencies 300, 200 and 600 having their means 16, 8 and 4 respectively, then the mean of combined distribution is
(a) 11
(b) 10
(c) 9
(d) 8
Answer: (d) 8

Question. When 10 is subtracted from all the observations, the mean is reduced to 60% of its value. If 5 is added to all the observations, then the mean will be
(a) 25
(b) 30
(c) 60
(d) 65
Answer: (b) 30

Question. A student has obtained 75%, 80% and 85% in three subjects. If the marks of another subject are added then his average can not be less than
(a) 60%
(b) 65%
(c) 80%
(d) 90%
Answer: (a) 60%

Question. The mean weight of 150 students in a certain class is 60 kilograms. The mean weight of boys in the class is 70 kilograms and that of the girls is 55 kilograms, then the number of boys and girls are
(a) 100,50
(b) 50,100
(c) 75,75
(d) 60, 90
Answer: (b) 50,100

Question. The following table given, the average score of the students is
No. of students (f) 8 12 20 10 6 4
Marks(x) 20 30 40 50 60 70
(a) 41
(b) 42
(c) 40
(d) 39
Answer: (a) 41

Question. The mean of following frequency table is 50.
Class 0-20 20-40 40-60 60-80 80-100 Total
Frequency 17 \( f_1 \) 32 \( f_2 \) 19 120
The missing frequencies are
(a) 28, 24
(b) 24, 36
(c) 36, 28
(d) 28, 34
Answer: (a) 28, 24

Question. Let \( G_1, G_2 \) be the geometric means of two series \( x_1, x_2, ......, x_n \); \( y_1, y_2, ......, y_n \). If G is the geometric mean of \( \frac{x_i}{y_i} \), i = 1, 2, ...... n, then G is equal to
(a) \( G_1 - G_2 \)
(b) \( \frac{\log G_1}{\log G_2} \)
(c) \( \frac{G_1}{G_2} \)
(d) \( G_1 + G_2 \)
Answer: (c) \( \frac{G_1}{G_2} \)

Question. If \( x_1, x_2, x_3 \) are three non zero real numbers such that \( (x_1^2 + x_2^2)(x_2^2 + x_3^2) \le (x_1 x_2 + x_2 x_3)^2 \) then the G.M. of \( x_1, x_2, x_3 \) is
(a) \( x_1 \)
(b) \( x_2 \)
(c) \( x_3 \)
(d) \( \frac{x_1 x_2 x_3}{3} \)
Answer: (b) \( x_2 \)

Question. The harmonic mean of the numbers 2, 3, 4 is
(a) 3
(b) \( \frac{1}{(24)^3} \)
(c) \( \frac{36}{13} \)
(d) \( \frac{13}{36} \)
Answer: (c) \( \frac{36}{13} \)

Question. If a variable taken discrete values \( x+4 \), \( x - \frac{7}{2} \), \( x - \frac{5}{2} \), x - 3, x - 2, \( x + \frac{1}{2} \), \( x - \frac{1}{2} \), x + 5 (x>0) then Median is
(a) \( x - \frac{5}{4} \)
(b) \( x - \frac{1}{2} \)
(c) x - 2
(d) \( x + \frac{5}{4} \)
Answer: (a) \( x - \frac{5}{4} \)

Question. The value of the mode given below is
Mark 0-10 10-20 20-30 30-40 40-50 50-60 60-70
Freq. 5 15 20 20 32 14 14
(a) 43
(b) 42
(c) 41
(d) 44
Answer: (d) 44

Question. Mean deviation of numbers 3, 4, 5, 6, 7 is
(a) 0
(b) 1.2
(c) 5
(d) 25
Answer: (b) 1.2

Question. The mean deviation of the following distribution is
x 10 11 12 13 14
f 3 12 18 12 3
(a) 12
(b) 0.75
(c) 1.25
(d) 26
Answer: (b) 0.75

Question. If \( \sum_{i=1}^{18} (x_i - 8) = 9 \) and \( \sum_{i=1}^{18} (x_i - 8)^2 = 45 \) then the standard deviation of \( x_1, x_2, ...., x_{18} \) is
(a) 4/9
(b) 9/4
(c) 3/2
(d) 2/3
Answer: (c) 3/2

Question. The mean square deviation of n observations \( x_1, x_2, ....., x_n \) about -2 and 2 are 18 and 10 respectively. Then S.D. of the given set is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. The median and S.D. of a distribution are 20 and 4 respectively. If each item is increased by 2, the new median and S.D. are
(a) 20, 6
(b) 22, 6
(c) 18, 6
(d) 22, 4
Answer: (d) 22, 4

Question. The mean of five observations is 4 and their variance is 5.2. If three of these observations are 1, 2, and 6. Then the other two are
(a) 2 and 9
(b) 3 and 8
(c) 4 and 7
(d) 5 and 6
Answer: (c) 4 and 7

Question. For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is
AIEEE - 2010
(a) 5/2
(b) 11/2
(c) 6
(d) 13/2
Answer: (b) 11/2

Question. The sum of squares of deviations for 10 items from the mean (=50) is 250. The coefficient of variation is
(a) 25
(b) 50
(c) 10
(d) None of the options
Answer: (c) 10

Question. If the standard deviation of 10 observations \( x_1, x_2, ......, x_{10} \) is 4 and that of another set of 10 observations \( y_1, y_2, ......, y_{10} \) is 3 and also \( X_i = (x_i - \bar{x})(y_i - \bar{y}) \), \( \bar{x} \) is mean of all \( x_i \)'s and \( \bar{y} \) is mean of all \( y_i \)'s . \( \sum_{i=1}^{10} X_i = 80 \) then standard deviation of ten observations \( (x_1 - y_1), (x_2 - y_2), ....., (x_{10} - y_{10}) \) is
(a) 1
(b) 3
(c) 5
(d) \( \sqrt{5} \)
Answer: (b) 3

ASCENDING & DESCENDING

Question. I. The geometric mean of 2, 4, 16 and 32 is a
II. The strength of 7 colleges in a city are 385, 1748, 1343, 1935, 786, 2874, 2108. Then the median strength is b.
III. The algebric sum of the deviations of 20 observations measured from 30 is 2. The mean of these observations is c.
(a) a < b < c
(b) b < c < a
(c) c < a < b
(d) a < c < b
Answer: (d) a < c < b

STATEMENTS

Question. Consider the following statements :
i) Mean of 100 observations is 50 and standard deviation is 10. If 5 is added to each observation the new mean and standard deviation are 55, 10.
ii) Mean of 100 observations is 50 and standard deviation is 10. If each observation is multipled by 3 then the new mean and standard deviation are \( 50, \frac{10}{3} \).
The true statements are :
(a) only (i)
(b) only (ii)
(c) both (i), (ii)
(d) neither (i) nor (ii)
Answer: (a) only (i)

MATCHING

Question. Match the correct parts to make a valid statement :
                List - I                                   List - II
A) Arithmetic Mean           1) \( l + [f_2 / (f_1 + f_2)] \times i \)
B) Geometric Mean          2) \( (x_1 \cdot x_2 \cdot ........ \cdot x_n)^{1/n} \)
C) Harmonic Mean           3) \( \sum fX / \sum f \)
D) Median                        4) \( l + \frac{N/2 - c.f}{f} \times i \)
E) Mode                            5) \( \left[ \frac{1}{n} \left( \frac{1}{x_1} + \frac{1}{x_2} + ...... + \frac{1}{x_n} \right) \right]^{-1} \)
                                         6) \( l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times i \)
(a) A B C D E - 3 5 4 1 2
(b) A B C D E - 2 4 1 5 3
(c) A B C D E - 3 2 5 4 6
(d) A B C D E - 1 3 2 4 6
Answer: (c) A B C D E - 3 2 5 4 6

Question. Match the values of \( Q_1, Q_2 \, \& \, Q_3 \) for the following data values
13, 14, 7, 12, 17, 8, 10, 6, 15, 18, 21, 20
        List - I             List - II
A) \( Q_1 \)             1) 13
B) \( Q_2 \)             2) 8.5
C) \( Q_3 \)             3) 17.5
(a) A B C - 1 2 3
(b) A B C - 2 1 3
(c) A B C - 3 2 1
(d) A B C - 3 1 2
Answer: (b) A B C - 2 1 3

ASSERTION & REASON

Question. Statement-I : The variance of first \( n \) even natural numbers is \( \frac{n^2 - 1}{4} \)
Statement-II : The sum of first \( n \) natural numbers is \( \frac{n(n + 1)}{2} \) and the sum of squares of first \( n \) natural numbers is \( \frac{n(n + 1)(2n + 1)}{6} \)
(AIEEE-2009)
(a) Statement-I is true, Statement-II is true ; Statement-II is not a correct explanation for Statement-I
(b) Statement-I is true, Statement-II is false
(c) Statement-I is false, Statement-II is true
(d) Statement-I is true, Statement-II is true, Statement-II is a correct explanation for Statement-I
Answer: (c) Statement-I is false, Statement-II is true

Question. Observe the following statements :
Statement-I : 10 is the mean of a set of 7 observations and 5 is the mean of a set of 3 observations. The mean of a combined set is 9.
Statement-II : If \( \bar{x}_i \quad (i = 1, 2, ....., k) \) are the means of k-series of sizes \( n_i \quad (i = 1, 2, 3, ....., k) \) respectively, then the combined or composite mean is \( \bar{x} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2 + ..... + n_k\bar{x}_k}{n_1 + n_2 + ..... + n_k} \)
(a) Statement-I is true, Statement-II is true, Statement-II is a correct explanation for Statement-I
(b) Statement-I is true, Statement-II is true; Statement-II is not a correct explanation for Statement-I
(c) Statement-I is true, Statement-II is false
(d) Statement-I is false, Statement-II is true
Answer: (d) Statement-I is false, Statement-II is true

Question. Let \( x_1, x_2, ....., x_n \) be n observations, and let \( \bar{x} \) be their arithmetic mean and \( \sigma^2 \) be their variance,
Statement I: Variance of \( 2x_1, 2x_2, ... 2x_n \) is \( 4\sigma^2 \)
Statement II: Arithmetic mean of \( 2x_1, 2x_2, ....., 2x_n \) is \( 4\bar{x} \) (AIEEE - 2012)
(a) Statement-I is true, Statement-II is true, Statement-II is a correct explanation for Statement-I
(b) Statement-I is true, Statement-II is true ; Statement-II is not a correct explanation for Statement-I
(c) Statement-I is true, Statement-II is false
(d) Statement-I is false, Statement-II is true
Answer: (c) Statement-I is true, Statement-II is false