Chapter 23

The average of \(n\) numbers \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\) is \(M .\) If \(x_{n}\) is replaced by \(x^{\prime}\), then new average is (A) \(M-x_{n}+x^{\prime}\) (B) \(\frac{n M-x_{n}+x^{\prime}}{n}\) (C) \(\frac{(n-1) M+x^{\prime}}{m}\) (D) \(\frac{M-x_{n}+x^{\prime}}{n}\)

The standard deviation of 25 numbers is \(40 .\) If each of the numbers is increased by 5, then the new standard deviation will be (A) 40 (B) 45 (C) \(40+\frac{21}{25}\) (D) None of these

The mean weight of 9 items is 51 . If one more item is added to the series the mean becomes 16 . The value of the 10 th item is (A) 35 (B) 30 (C) 25 (D) 20

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 of 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

If a variable \(x\) takes values \(0,1,2, \ldots, n\) with frequencies proportional to the binomial coefficients \({ }^{n} C_{0}\), \({ }^{n} C_{1},{ }^{n} C_{2}, \ldots,{ }^{n} C_{n}\), then the Var \((x)\) is (A) \(\frac{n^{2}-1}{12}\) (B) \(\frac{n}{2}\) (C) \(\frac{n}{4}\) (D) None of these

The sum of squares of deviations for 10 observations taken from mean 50 is 250 . The coefficient of variation is (A) \(50 \%\) (B) \(10 \%\) (C) \(40 \%\) (D) None of these

If the standard deviation of \(n\) observations \(x_{1}, x_{2}, \ldots\), \(x_{n}\) is 4 and another set of \(n\) observations \(y_{1}, y_{2}, \ldots, y_{n}\) is 3 . The standard deviation of \(n\) observations \(x_{1}-y_{1}\), \(x_{2}-y_{2}, \ldots, x_{n}-y_{n}\) is (A) 1 (B) \(\frac{2}{\sqrt{3}}\) (C) 5 (D) Data insufficient

Let \(r\) be the range and \(S^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}\) be the S.D. of a set of observations \(x_{1}, x_{2}, \ldots, x_{n}\), then (A) \(S \leq r \sqrt{\frac{n}{n-1}}\) (B) \(S=r \sqrt{\frac{n}{n-1}}\) (C) \(S \geq r \sqrt{\frac{n}{n-1}}\) (D) None of these

The A.M. of \(n\) numbers of a series is \(\bar{x} .\) If the sum of the first \((n-1)\) term is \(k\), them the \(n\)th number is (A) \(\bar{x}-k\) (B) \(n \bar{x}-k\) (B) \(\bar{x}-n k\) (D) \(n \bar{x}-n k\)

If a variable takes values \(0,1,2, \ldots, n\) with frequencies \(q^{n}, \frac{n}{1} q^{n-1} p, \frac{n(n-1)}{1.2} q^{n-2} p^{2}, \ldots, p^{n}\), where \(p+q=1\) then the mean is (A) \(n p\) (B) \(n q\) (C) \(n(p+q)\) (D) None of these

The S.D. of a variate \(x\) is \(\sigma .\) The S.D. of the variate \(\frac{a x+b}{c}\) where \(a, b, c\) are constants, is (A) \(\left(\frac{a}{c}\right) \sigma\) (B) \(\left|\frac{a}{c}\right| \sigma\) (C) \(\left(\frac{a^{2}}{c^{2}}\right) \sigma\) (D) None of these

Consider any set of observations \(x_{1}, x_{2}, x_{3}, \ldots, x_{101} ;\) it being given that \(x_{1}

The mean of the numbers \(\frac{{ }^{50} C_{0}}{1}, \frac{{ }^{50} C_{2}}{3}, \frac{{ }^{50} C_{4}}{5} \ldots, \frac{{ }^{50} C_{50}}{51}\) equals (A) \(\frac{2^{50}}{51}\) (B) \(\frac{2^{49}}{51}\) (C) \(\frac{2^{49}}{39 \times 17}\) (D) None of these

The standard deviation of a distribution is 30 and each item is raised by 3 , then new S.D. is (A) 32 (B) 28 (C) 27 (D) None of these

For three numbers \(a, b, c\) product of the average of the numbers \(a^{2}, b^{2}, c^{2}\) and \(\frac{1}{a^{2}}, \frac{1}{b^{2}}, \frac{1}{c^{2}}\) cannot be less than (A) 1 (B) 3 (C) 9 (D) None of these

The variance of \(\alpha, \beta\) and \(\gamma\) is 9 , then variance of \(5 \alpha, 5 \beta\) and \(5 \gamma\) is (A) 45 (B) \(9 / 5\) (C) \(5 / 9\) (D) 225

Mean of the numbers \(1,2,3, \ldots, n\) with respective weights \(1^{2}+1,2^{2}+2,3^{2}+3, \ldots, n^{2}+n\) is (A) \(\frac{3 n(n+1)}{2(2 n+1)}\) (B) \(\frac{2 n+1}{3}\) (C) \(\frac{3 n+1}{4}\) (D) \(\frac{3 n+1}{2}\)

The G.M. of the number \(3,3^{2}, 3^{3}, \ldots, 3^{3 n}\) is (A) \(3^{\frac{n}{2}}\) (B) \(3^{\frac{3 n}{2}}\) (C) \(3^{\frac{3 n+1}{2}}\) (D) \(3^{\frac{n+1}{2}}\)

The reciprocal of the weighted mean of first \(n\) natural numbers whose weights are equal to the squares of the corresponding numbers is (A) \(\frac{2(2 n+1)}{3 n(n+1)}\) (B) \(\frac{3 n(n+1)}{n(2 n+1)}\) (C) \(\frac{3 n(n+1)}{2 n+1}\) (D) None of these

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

The mean weight per student in a group of seven students is \(55 \mathrm{~kg}\) If the individual weights of 6 students are \(52,58,55,53,56\) and 54 ; then weight of the seventh student is (A) \(55 \mathrm{~kg}\) (B) \(60 \mathrm{~kg}\) (C) \(57 \mathrm{~kg}\) (D) \(50 \mathrm{~kg}\)

If the mean of a set of observations \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\) is \(\bar{x}\), then mean of observations \(x_{i}+3 i \forall i=1,2,3, \ldots n\) equals (A) \(\bar{x}+3(n+1)\) (B) \(\bar{x}+\frac{3(n+1)}{2}\) (C) \(\bar{x}+\frac{n+1}{2 n}\) (D) None of these

The weighted mean of the square of 1st \(n\) natural numbers whose weights are corresponding numbers, equals (A) \(\frac{(n+1)(2 n+1)}{2}\) (B) \(\frac{n(n+1)}{2}\) (C) \(\frac{n+1}{2}\) (D) None of these

If the variate of a distribution takes the values 1,2 , \(3, \ldots n\) with frequencies \(n, n-1, n-2, \ldots .3,2,1\), then mean value of the distribution is (A) \(\frac{n(n+2)}{3}\) (B) \(\frac{n(n+1)(n+2)}{6}\) (C) \(\frac{n+2}{3}\) (D) \(\frac{(n+1))(n+2)}{6}\)

The means 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

The mean of \(n\) items is \(\bar{x}\). If each item is successively increased by \(3,3^{2}, 3^{3}, \ldots .3^{n}\), then new mean equals (A) \(\bar{x}+\frac{3^{n+1}}{n}\) (B) \(\bar{x}+3 \frac{\left(3^{n}-1\right)}{2 n}\) (C) \(\bar{x}+\frac{3^{n}}{n}\) (D) \(\bar{x}+3 \frac{\left(3^{n}-1\right)}{2 n}\)

The average salary of male employees in a firm was Rs. 520 and that of females was Rs. 420 . The mean salary of all the employees was Rs. \(500 .\) The percentage of male employees is (A) 80 (B) 60 (B) 40 (D) 20

The average weight of students in a class of 35 students is \(40 \mathrm{~kg}\). If the weight of the teacher be included, the average rises by \(\frac{1}{2} \mathrm{~kg}\); the weight of the teacher is (A) \(40.5 \mathrm{~kg}\) (B) \(50 \mathrm{~kg}\) (C) \(41 \mathrm{~kg}\) (D) \(58 \mathrm{~kg}\)

An automobile driver travels from plane to a hill station \(120 \mathrm{~km}\) distant at an average speed of \(30 \mathrm{~km}\) per hour. He then makes the return trip at an average speed of \(25 \mathrm{~km}\) per hour. He covers another \(120 \mathrm{~km}\) distance on plane at an average speed of \(50 \mathrm{~km}\) per hour. His average speed over the entire distance of \(360 \mathrm{~km}\) will be (A) \(\frac{30+25+50}{3} \mathrm{~km} / \mathrm{h}\) (B) \((30 \cdot 25 \cdot 50)^{\frac{1}{3}}\) (C) \(\frac{3}{\frac{1}{30}+\frac{1}{25}+\frac{1}{50}} \mathrm{~km} / \mathrm{h}\) (D) None of these

If the mean deviation about the median of the numbers \(a, 2 a, \ldots, 50 a\) is 50 , then \(|a|\) equals (A) 5 (B) 2 (C) 3 (D) 4

Let \(x_{1}, x_{2}, \ldots, x_{n}\) be \(n\) observations, and let \(\bar{x}\) be their arithmetic mean and \(\sigma^{2}\) be the variance Statement-1: Variance of \(2 x_{1}, 2 x_{2}, \ldots, 2 x_{n}\) is \(4 \sigma^{2}\). Statement-2: Arithmetic mean \(2 x_{1}, 2 x_{2}, \ldots, 2 x_{n}\) is \(4 \bar{x}\). (A) Statement- 1 is false, Statement- 2 is true. (B) Statement-1 is true, statement- 2 is true; statement- 2 is a correct explanation for Statement-1. (C) Statement-1 is true, statement-2 is true; statement- 2 is not a correct explanation for Statement-1. (D) Statement- 1 is true, statement- 2 is false.

If the mean of a set of observations \(x_{1}, x_{2}, \ldots, x_{10}\) is 20 then the mean of \(x_{1}+4, x_{2}+8, x_{3}+12, \ldots, x_{10}+40\) is (A) 34 (B) 42 (C) 38 (D) 40

The mean of the numbers \(a, b, 8,5,10\) is 6 and the variance is \(6.80 .\) Then which one of the following gives possible values of \(a\) and \(b\) ? (A) \(a=0, b=7\) (B) \(a=5, b=2\) (C) \(a=1, b=6\) (D) \(a=3, b=4\)

If the mean deviation of number \(1,1+d, 1+2 d, \ldots, 1\) \(+100 d\) from their mean is 255 , then the \(d\) is equal to (A) \(10.0\) (B) \(20.0\) (C) \(10.1\) (D) \(20.2\)

Statement-1: The variance of first \(n\) even natural numbers is \(\frac{n^{2}-1}{4}\) Statement-2: The sum of first \(n\) natural numbers is \(\frac{n(n+1)}{2}\) and the sum of squares of first \(n\) natural numbers is \(\frac{n(n+1)(2 n+1)}{6}\) (A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1. (B) Statement- 1 is true, Statement- 2 is true; Statement- 2 is not a correct explanation for Statement-1. (C) Statement- 1 is true, Statement- 2 is false. (D) Statement- 1 is false, Statement- 2 is true.

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 [2003] (A) is increased by 2 (B) is decreased by 2 (C) is two times the original median (D) remains the same as that of the original set

Let two numbers have arithmetic mean 9 and geometric mean 4 . Then these numbers are the roots of the quadratic equation [2004] (A) \(x^{2}+18 x+16=0\) (B) \(x^{2}-18 x-16=0\) (C) \(x^{2}+18 x-16=0\) (D) \(x^{2}-18 x+16=0\)

In a series of \(2 n\) observations, half of them equal \(a\) and remaining half equal \(-a\). If the standard deviation of the observations is 2 , then \(|a|\) equals (A) \(\frac{1}{n}\) (B) \(\sqrt{2}\) (C) 2 (D) \(\frac{\sqrt{2}}{n}\)

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) \(24.0\)

Let \(x_{1}, x 2, \ldots, x_{n}\) be \(n\) observations such that \(\sum x_{i}^{2}=400\) and \(\sum x_{i}=80 .\) Then a possible value of \(n\) among the following is [2005] (A) 15 (B) 18 (C) 9 (D) 12

Suppose a population \(A\) has 100 observations 101 , \(102, \ldots, 200\), and another population \(B\) has 100 observations \(151,152, \ldots, 250\). If \(V_{A}\) and \(V_{B}\) represent the variances of the two populations, respectively, then \(\frac{V_{A}}{V_{B}}\) is [2006] (A) 1 (B) \(9 / 4\) (C) \(4 / 9\) (D) \(2 / 3\)

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 [2007] (A) 40 (B) 20 (C) 80 (D) 60

The mean of the numbers \(a, b, 8,5,10\) is 6 and the variance is \(6.80 .\) Then which one of the following gives possible values of \(a\) and \(b ?\) (A) \(a=0, b=7\) (B) \(a=5, b=2\) (C) \(a=1, b=6\) (D) \(a=3, b=4\)

If the mean deviation of number \(1,1+d, 1+2 d, \ldots \ldots\), \(1+100 \mathrm{~d}\) from their mean is 255 , then the \(\mathrm{d}\) is equal to [2009] (A) \(10.0\) (B) \(20.0\) (C) \(10.1\) (D) \(20.2\)

For two data sets, each with 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 (A) \(\frac{11}{2}\) (B) 6 (C) \(\frac{13}{2}\) (D) \(\frac{5}{2}\)

If the mean deviation about the median of the numbers \(a, 2 a \ldots .50 a\) is 50, then \(|a|\) equals (A) 3 (B) 4 (C) 5 (D) 2

Let \(x_{1}, x_{2} \ldots x_{n}\) be \(n\) observations, and let \(\bar{x}\) be their arithmetic mean and \(\sigma^{2}\) be their variance. Statement-1: Variance of \(2 x_{1}, 2 x_{2} \ldots 2 x_{n}\) is \(4 \sigma^{2}\). Statement-2: Arithmetic mean of \(2 x_{1}, 2 x_{2} \ldots 2 x_{n}\) is 4 \(\bar{x} .\) (A) Statement-1 is false, statement- 2 is true (B) Statement- 1 is true, statement- 2 is true; statement 2 is a correct explanation for statement 1 (C) Statement- 1 is true, statement-2 is true; statement 2 is not a correct explanation for statement 1 (D) Statement-1 is true, statement- 2 is false

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to entire class. Which of the following statistical measures will not change even after the grace marks were given? [2013] (A) median (B) mode (C) variance (D) mean

The variance of the first 50 even natural numbers is [2014] (A) \(\frac{833}{4}\) (B) 833 (C) 437 (D) \(\frac{437}{4}\)

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 [2015] (A) \(16.0\) (B) \(15.8\) (C) \(14.0\) (D) \(16.8\)