Chapter 14

Consider the data on \(x\) taking the values \(0,2,4,8, \ldots, 2^{n}\) with frequencies \({ }^{n} \mathrm{C}_{0}{\underline{\phantom{xx}}}^{n} \mathrm{C}_{1},{ }^{n} \mathrm{C}_{2}, \ldots,{ }^{n} \mathrm{C}_{\mathrm{n}}\) respectively. If the mean of this data is \(\frac{728}{2^{n}}\), then \(n\) is equal to

The minimum value of \(2^{\sin x}+2^{\cos x}\) is: \([\) Sep. 04, \(\mathbf{2 0 2 0}\) (II)](a) \(\begin{array}{llll}2^{-1+\frac{1}{\sqrt{2}}} & \text { (b) } 2^{-1+\sqrt{2}} & \text { (c) } 2^{1-\sqrt{2}} & \text { (d) } 2^{1-\frac{1}{\sqrt{2}}}\end{array}\)

The mean and the median of the following ten numbers in increasing order \(10,22,26,29,34, x, 42,67,70, y\) are 42 and 35 respectively, then \(\frac{y}{x}\) is equal to: [April. 09, 2019 (II)] \(x\) (a) \(9 / 4\) (b) \(7 / 2\) (c) \(8 / 3\) (d) \(7 / 3\)

The mean of a set of 30 observations is 75 . If each other observation is multiplied by a non-zero number \(\lambda\) and then each of them is decreased by 25 , their mean remains the same. The \(\lambda\) is equal to [Online April 15, 2018] (a) \(\frac{10}{3}\) (b) \(\frac{4}{3}\) (c) \(\frac{1}{3}\) (d) \(\frac{2}{3}\)

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If now the mean age of the teachers in this school is 39 years, then the age (in years) of the newly appointed teacher is : [Online April 8, 2017] (a) 25 (b) 30 (c) 35 (d) 40

The mean of the data set comprising of 16 observations is 16\. If one of the observation valued 16 is deleted and three new observations valued 3,4 and 5 are added to the data, then the mean of the resultant data, is: (a) \(15.8\) (b) \(14.0\) (c) \(16.8\) (d) \(16.0\)

Let the sum of the first three terms of an A. P, be 39 and the sum of its last four terms be 178 . If the first term of this A.P. is 10 , then the median of the A.P. is : [Online April 10, 2015] (a) 28 (b) \(26.5\) (c) \(29.5\) (d) 31

Let the sum of the first three terms of an A. P, be 39 and the sum of its last four terms be 178 . If the first term of this A.P. is 10 , then the median of the A.P. is : [Online April 10, 2015] (a) 28 (b) \(26.5\) (c) \(29.5\) (d) 31

In a set of \(2 \mathrm{n}\) distinct observations, each of the observations below the median of all the observations is increased by 5 and each of the remaining observations is decreased by \(3 .\) Then the mean of the new set of observations: \(\quad[\) Online April 9, 2014] (a) increases by 1 (b) decreases by 1 (c) decreases by 2 (d) increases by 2

If the median and the range of four numbers \(\\{x, y, 2 x+y, x-y\\}\), where \(0

The mean of a data set consisting of 20 observations is 40 . If one observation 53 was wrongly recorded as 33 , then the correct mean will be : \(\quad\) [Online April9, 2013] (a) 41 (b) 49 (c) \(40.5\) (d) \(42.5\)

The median of 100 observations grouped in classes of equal width is 25 . If the median class interval is \(20-30\) and the number of observations less than 20 is 45 , then the frequency of median class is [Online May 19, 2012] (a) 10 (b) 20 (c) 15 (d) 12

The average marks of boys in 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 [2007] (a) 80 (b) 60 (c) 40 (d) 20

Let \(x_{1}, x_{2}, \ldots \ldots \ldots \ldots \ldots, x_{n}\) be \(\mathrm{n}\) observations such that \(\sum x_{i}^{2}=400\) and \(\sum x_{i}=80 .\) Then the possible value of \(n\) among the following is (a) 15 (b) 18 (c) 9 (d) 12

If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately [2005] (a) \(22.0\) (b) \(20.5\) (c) \(25.5\) (d) \(24.0\)

The median of a set of 9 distinct observations is \(20.5\). If each of the largest 4 observations of the set is increased by 2, then the median of the new set (a) remains the same as that of the original set (b) is increased by 2 (c) is decreased by 2 (d) is two times the original median.

In a class of 100 students there are 70 boys whose average marks in a subject are 75 . If the average marks of the complete class is 72 , then what is the average of the girls? [2002] (a) 73 (b) 65 (c) 68 (d) 74

If \(\sum_{i=1}^{n}\left(x_{i}-a\right)=n\) and \(\sum_{i=1}^{n}\left(x_{i}-a\right)^{2}=n a,(n, a>1)\), then the standard deviation of \(n\) observations \(x_{1}, x_{2}, \ldots, x_{n}\) is: [Sep. 06, 2020 (I)] (a) \(a-1\) (b) \(n \sqrt{a-1}\) (c) \(\sqrt{n(a-1)}\) (d) \(\sqrt{a-1}\)

The mean and variance of 7 observations are 8 and 16 , respectively. If five observations are \(2,4,10,12,14\), then the absolute difference of the remaining two observations is: (a) 1 (b) 4 (c) 2 (d) 3

If the mean and the standard deviation of the data \(3,5,7, a\), \(b\) are 5 and 2 respectively, then \(a\) and \(b\) are the roots of the equation: [Sep. 05, 2020 (II)] (a) \(x^{2}-10 x+18=0\) (b) \(2 x^{2}-20 x+19=0\) (c) \(x^{2}-10 x+19=0\) (d) \(x^{2}-20 x+18=0\)

The mean and variance of 8 observations are 10 and \(13.5\), respectively. If 6 of these observations are \(5,7,10,12,14\), 15 , then the absolute difference of the remaining two observations is : (a) 9 (b) 5 (c) 3 (d) 7

For the frequency distribution :where \(00\), the standard deviation cannot be: \(\quad\) [Sep. 03, 2020 (I)] (a) 4 (b) 1 (c) 6 (d) 2

Let \(x_{i}(1 \leq i \leq 10)\) be ten observations of a random variable \(X\). If \(\sum_{i=1}^{10}\left(x_{i}-p\right)=3\) and \(\sum_{i=1}^{10}\left(x_{i}-p\right)^{2}=9\) where \(0 \neq p \in \mathbf{R}\), then the standard deviation of these observations is: \(\quad\) [Sep. 03, 2020 (II)] (a) \(\sqrt{\frac{3}{5}}\) (b) \(\frac{4}{5}\) (c) \(\frac{9}{10}\) (d) \(\frac{7}{10}\)

Let \(X=\\{x \in \mathbf{N}: 1 \leq x \leq 17\\}\) and \(Y=\\{a x+b: x \in X\) and \(a, b \in \mathbf{R}, a>0\\} .\) If mean and variance of elements of \(Y\) are 17 and 216 respectively then \(a+b\) is equal to : [Sep. \(\mathbf{0 2}, \mathbf{2 0 2 0}(\mathbf{I})]\) (a) 7 (b) \(-7\) (c) \(-27\) (d) 9

If the variance of the terms in an increasing A.P., \(b_{1}, b_{2}, b_{3}, \ldots ., b_{11}\) is 90 , then the common difference of this A.P. is [NA Sep. 02, 2020 (II)]

Let the observations \(x_{i}(1 \leq i \leq 10)\) satisfy the equations, \(\sum_{i=1}^{10}\left(x_{i}-5\right)=10\) and \(\sum_{i=1}^{10}\left(x_{i}-5\right)^{2}=40 .\) If \(\mu\) and \(\lambda\) are the mean and the variance of the observations, \(x_{1}-3, x_{2}-3\), \(\ldots, x_{10}-3\), then the ordered pair \((\mu, \lambda)\) is equal to: [Jan. 9, 2020 (I)] (a) \((3,3)\) (b) \((6,3)\) (c) \((6,6)\) (d) \((3,6)\)

The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by \(p\) and then reduced by \(q\), where \(p \neq 0\) and \(q \neq 0 .\) If the new mean and new s.d. become half of their original values, then \(q\) is equal to: [Jan. 8, 2020 (I)] (a) \(-5\) (b) 10 (c) \(-20\) (d) \(-10\)

The mean and variance of 20 observations are found to be 10 and 4 , respectively. On rechecking, it was found that an observation 9 was incorrect and the correct observation was 11 . Then the correct variance is: [Jan. 8, 2020 (II)] (a) \(3.99\) (b) \(4.01\) (c) \(4.02\) (d) \(3.98\)

If the variance of the first \(n\) natural numbers is 10 and the variance of the first \(m\) even natural numbers is 16 , then \(m+n\) is equal to \(-\) [NA Jan. 7, 2020 (I)]

If the mean and variance of eight numbers \(3,7,9,12,13\), \(20, x\) and \(y\) be 10 and 25 respectively, then \(x \cdot y\) is equal to [NA Jan. 7, 2020 (II)]

If the data \(x_{1}, x_{2}, \ldots \ldots, x_{10}\) is such that the mean of first four of these is 11 , the mean of the remaining six is 16 and the sum of squares of all of these is 2,000 ; then the standard deviation of this data is : \(\quad\) [April 12, 2019 (I)] (a) \(2 \sqrt{2}\) (b) 2 (c) 4 (d) \(\sqrt{2}\)

If both the mean and the standard deviation of 50 observations \(x_{1}, x_{2}, \ldots ., x_{50}\) are equal to 16, then the mean of \(\left(x_{1}-4\right)^{2},\left(x_{2}-4\right)^{2}, \ldots,\left(x_{50}-4\right)^{2}\) is: \(\quad\) [April 10, \(\mathbf{2 0 1 9}\) (II)] (a) 400 (b) 380 (c) 525 (d) 480

If the standard deviation of the numbers \(-1,0,1, \mathrm{k}\) is \(\sqrt{5}\) where \(\mathrm{k}>0\), then \(\mathrm{k}\) is equal to: \(\quad\) [April09, \(\mathbf{2 0 1 9}\) (I)] (a) \(2 \sqrt{6}\) (b) \(2 \sqrt{\frac{10}{3}}\) (c) \(4 \sqrt{\frac{5}{3}}\) (d) \(\sqrt{6}\)

The mean and variance of seven observations are 8 and 16, respectively. If 5 of the observations are \(2,4,10,12,14\), then the product of the remaining two observations is : \([\) April \(08,2019(\mathrm{I})]\) (a) 45 (b) 49 (c) 48 (d) 40

A student scores the following marks in five tests: 45,54 , \(41,57,43\). His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is : [April. \(\mathbf{0 8}, \mathbf{2 0 1 9}\) (II)] (a) \(\frac{10}{\sqrt{3}}\) (b) \(\frac{100}{3}\) (c) \(\frac{10}{3}\) (d) \(\frac{100}{\sqrt{3}}\)

If the sum of the deviations of 50 observations from 30 is 50 , then the mean of these observations is : [Jan. 12, 2019 (I)] (a) 30 (b) 51 (c) 50 (d) 31

The mean and the variance of five observations are 4 and \(5.20\), respectively. If three of the observations are 3,4 and 4; then the absolute value of the difference of the other two observations, is: \(\quad\) [Jan. 12, 2019 (II)] (a) 7 (b) 5 (c) 1 (d) 3

The outcome of each of 30 items was observed; 10 items gave an outcome \(\frac{1}{2}-\mathrm{d}\) each, 10 items gave outcome \(\frac{1}{2}\) each and the remaining 10 items gave outcome \(\frac{1}{2}+\mathrm{d}\) each. If the variance of this outcome data is \(\frac{4}{3}\) then \(|\mathrm{d}|\) equals : [Jan. 11, 2019 (I)] (a) \(\frac{2}{3}\) (b) 2 (c) \(\frac{\sqrt{5}}{2}\) (d) \(\sqrt{2}\)

A data consists of \(\mathrm{n}\) observations: \(x_{1}, x_{2}, \ldots, x_{n} .\) If \(\sum_{i=1}^{\mathrm{n}}\left(x_{i}+1\right)^{2}=9 \mathrm{n}\) and \(\sum_{i=1}^{\mathrm{n}}\left(x_{i}-1\right)^{2}=5 \mathrm{n}\) then the standard deviation of this data is: [Jan. 09, 2019 (II)] (a) 2 (b) \(\sqrt{5}\) (c) 5 (d) \(\sqrt{7}\)

The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1,3 and 8 , then a ratio of other two observations is: [Jan. 10, 2019 (I)] (a) \(10: 3\) (b) \(4: 9\) (c) \(5: 8\) (d) \(6: 7\)

If mean and standard deviation of 5 observations \(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\) are 10 and 3, respectively, then the variance of 6 observations \(x_{1}, x_{2}, \ldots, x_{5}\) and \(-50\) is equal to: [Jan. 10, 2019 (II)] (a) \(509.5\) (b) \(586.5\) (c) \(582.5\) (d) \(507.5\)

5 students of a class have an average height \(150 \mathrm{~cm}\) and variance \(18 \mathrm{~cm}^{2}\). A new student, whose height is \(156 \mathrm{~cm}\), joined them. The variance (in \(\mathrm{cm}^{2}\) ) of the height of these six students is: [Jan. 9, 2019 (I)] (a) 16 (b) 22 (c) 20 (d) 18

The mean and the standard deviation (s.d.) of five observations are 9 and 0 , respectively. If one of the observations is changed such that the mean of the new set of five observations becomes 10 , then their s.d. is? \(\quad\) [Online April 16, 2018] (a) 0 (b) 4 (c) 2 (d) 1

If the mean of the data \(: 7,8,9,7,8,7, \lambda, 8\) is 8, then the variance of this data is \(\quad\) [Online April 15, 2018] (a) \(\frac{9}{8}\) (b) 2 (c) \(\frac{7}{8}\) (d) 1

If \(\sum_{\mathrm{i}=1}^{9}\left(\mathrm{x}_{\mathrm{i}}-5\right)=9\) and \(\sum_{\mathrm{i}=1}^{9}\left(\mathrm{x}_{\mathrm{i}}-5\right)^{2}=45\), then the standard deviation of the 9 items \(x_{1}, x_{2}, \ldots, x_{9}\) is: [2018] (a) 4 (b) 2 (c) 3 (d) 9

The sum of 100 observations and the sum of their squares are 400 and 2475, respectively. Later on, three observations, 3,4 and 5, were found to be incorrect. If the incorrect observations are omitted, then the variance of the remaining observations is : \(\quad\) [Online April 9, 2017] (a) \(8.25\) (b) \(8.50\) (c) \(8.00\) (d) \(9.00\)

If the standard deviation of the numbers \(2,3, \mathrm{a}\) and 11 is 3.5, then which of the following is true? (a) \(3 \mathrm{a}^{2}-34 \mathrm{a}+91=0\) (b) \(3 a^{2}-23 a+44=0\) (c) \(3 \mathrm{a}^{2}-26 \mathrm{a}+55=0\) (d) \(3 \mathrm{a}^{2}-32 \mathrm{a}+84=0\)

The mean of 5 observations is 5 and their variance is 124 . If three of the observations are 1,2 and 6 ; then the mean deviation from the mean of the data is : [Online April 10, 2016] (a) \(2.5\) (b) \(2.6\) (c) \(2.8\) (d) \(2.4\)

If the mean deviation of the numbers \(1,1+\mathrm{d}, \ldots, 1+100 \mathrm{~d}\) from their mean is 255 , then a value of \(\mathrm{d}\) is : [Online April 9, 2016] (a) \(10.1\) (b) \(5.05\) (c) \(20.2\) (d) 10

The variance of first 50 even natural numbers is [2014] (a) 437 (b) \(\frac{437}{4}\) (c) \(\frac{833}{4}\) (d) 833