Chapter 14

Let \(\overline{\mathrm{x}}, \mathrm{M}\) and \(\sigma^{2}\) be respectively the mean, mode and variance of \(n\) observations \(x_{1}, x_{2}, \ldots, x_{n}\) and \(d_{i}=-x_{i}-a\), \(\mathrm{i}=1,2, \ldots, \mathrm{n}\), where \(\mathrm{a}\) is any number. Statement I: Variance of \(\mathrm{d}_{1}, \mathrm{~d}_{2}, \ldots \mathrm{d}_{\mathrm{n}}\) is \(\sigma^{2}\). Statement II: Mean and mode of \(\mathrm{d}_{1}, \mathrm{~d}_{2}, \ldots . \mathrm{d}_{\mathrm{n}}\) are \(-\overline{\mathrm{x}}-\mathrm{a}\) and \(-\mathrm{M}-\mathrm{a}\), respectively. [Online April 19, 2014] (a) Statement I and Statement II are both false (b) Statement I and Statement II are both true (c) Statement Iis true and Statement II is false (d) Statement I is false and Statement II is true

Let \(\bar{X}\) and M.D. be the mean and the mean deviation about \(\overline{\mathrm{X}}\) of n observations \(\mathrm{x}_{\mathrm{i}}, \mathrm{i}=1,2, \ldots \ldots, \mathrm{n}\). If each of the observations is increased by 5 , then the new mean and the mean deviation about the new mean, respectively, are: [Online April 12, 2014] (a) \(\overline{\mathrm{X}}, \mathrm{M} . \mathrm{D}\). (b) \(\overline{\mathrm{X}}+5, \mathrm{M} . \mathrm{D}\). (c) \(\overline{\mathrm{X}}, \mathrm{M} . \mathrm{D} .+5\) (d) \(\overline{\mathrm{X}}+5, \mathrm{M} . \mathrm{D} .+5\)

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) mean (b) median (c) mode (d) variance

In a set of 2 n observations, half of them are equal to 'a' and the remaining half are equal to '-a'. If the standard deviation of all the observations is 2 ; then the value of \(|a|\) is : [Online April 25, 2013] (a) 2 (b) \(\sqrt{2}\) (c) 4 (d) \(2 \sqrt{2}\)

Mean of 5 observations is 7 . If four of these observations are \(6,7,8,10\) and one is missing then the variance of all the five observations is : [Online April 22, 2013] (a) 4 (b) 6 (c) 8 (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 the variance. Statement-1: Variance of \(2 x_{1}, 2 x_{2}, \ldots, 2 x_{n}\) is \(4 \mathrm{o}^{2}\). Statement- \(\mathbf{2}\) : Arithmeticmean \(2 x_{1}, 2 x_{2}, \ldots, 2 x_{n}\) is \(4 \bar{x}\). \([2012]\) (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.

Statement 1 : The variance of first \(n\) odd natural numbers is \(\frac{n^{2}-1}{3}\) Statement 2: The sum of first \(\mathrm{n}\) odd natural number is \(n^{2}\) and the sum of square of first \(n\) odd natural numbers is \(\frac{n\left(4 n^{2}+1\right)}{3}\). Online May 26, 2012] (a) Statement 1 is true, Statement 2 is false. (b) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement \(1 .\) (c) Statement 1 is false, Statement 2 is true. (d) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1 .

If the mean of \(4,7,2,8,6\) and a is 7 , then the mean deviation from the median of these observations is [Online May 12, 2012] (a) 8 (b) 5 (c) 1 (d) 3

A scientist is weighing each of 30 fishes. Their mean weight worked out is \(30 \mathrm{gm}\) and a standarion deviation of \(2 \mathrm{gm}\). Later, it was found that the measuring scale was misaligned and always under reported every fish weight by \(2 \mathrm{gm}\). The correct mean and standard deviation (in gm) of fishes are respectively: (a) 32,2 (b) 32,4 (c) 28,2 (d) 28,4

If the mean deviation about the median of the numbers \(a\), \(2 a, \ldots \ldots, 50 a\) is 50 , then \(|a|\) equals [2011] (a) 3 (b) 4 (c) 5 (d) 2

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 [2010] (a) \(\frac{11}{2}\) (b) 6 (c) \(\frac{13}{2}\) (d) \(\frac{5}{2}\)

If the mean deviation of the numbers \(1,1+\mathrm{d}\), \(1+2 \mathrm{~d}, \ldots .1+100 d\) from their mean is 255 , then \(d\) is equal to: [2009] (a) \(20.0\) (b) \(10.1\) (c) \(20.2\) (d) \(10.0\)

Statement-1 : The variance of first \(\mathrm{n}\) 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 \(\frac{n(n+1)(2 n+1)}{6}\). [2009] (a) Statement- 1 is true, Statement- 2 is true. Statement- 2 is not a correct explanation for Statement-1. (b) Statement- 1 is true, Statement- 2 is false.

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) \(a=3, b=4\)

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

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) \(\frac{\sqrt{2}}{n}\) (b) \(\sqrt{2}\) (c) 2 (d) \(\frac{1}{n}\)

Consider the following statements : (A) Mode can be computed from histogram (B) Median is not independent of change of scale (C) Variance is independent of change of origin and scale. Which of these is / are correct? (a) (A), (B) and (C) (b) Only(B) (c) Only (A) and (B) (d) Only (A)

In an experiment with 15 observations on \(x\), the following results were available: \(\Sigma x^{2}=2830, \Sigma x=170\) One observation that was 20 was found to be wrong and was replaced by the correct value 30. The corrected variance is (a) \(8.33\) (b) \(78.00\) (c) \(188.66\) (d) \(177.33\)