Statistics · JEE Main previous year questions

A set of four observations has mean 1 and variance 13. Another set of six observations has mean 2 and variance 1. Then, the variance of all these 10 observations is equal to :

1. A.$5.96$
2. B.$6.14$
3. C.$6.04$
4. D.$6.24$

A data consists of 20 observations $x_1,x_2,\dots ,x_{20}$. If ${\sum }_{i=1}^{20}(x_i+5{)}^2=2500$ and ${\sum }_{i=1}^{20}(x_i-5{)}^2=100$, then the ratio of mean to standard deviation of this data is:

1. A.2:1
2. B.3:1
3. C.3:2
4. D.4:1

1. A.$x^2-35x+306=0$
2. B.$x^2-41x+420=0$
3. C.$x^2-45x+506=0$
4. D.$x^2-37x+342=0$

The mean deviation about the mean for the data $$\begin{array}{ccccccc}{x}_i& 5& 7& 9& 10& 12& 15\\ f_i& 8& 6& 2& 2& 2& 6\end{array}$$ is equal to:

1. A.$\frac{40}{13}$
2. B.$\frac{42}{13}$
3. C.$\frac{44}{13}$
4. D.$\frac{46}{13}$

A variable $\mathrm{X}$ takes values $0,0,2,6,12,20,\dots ,n(n-1)$ with frequencies ${}^n{\mathrm{C}}_0,{}^n{\mathrm{C}}_1,{}^n{\mathrm{C}}_2,{}^n{\mathrm{C}}_3,{}^n{\mathrm{C}}_4,{}^n{\mathrm{C}}_5,\dots ,{}^n{\mathrm{C}}_n$ respectively. If the mean of this data is $60$, then its median is :

1. A.$56$
2. B.$42$
3. C.$72$
4. D.$90$

Suppose that the mean and median of the non-negative numbers $21,8,17,a,51,103,b,13,67$, ($a>b$), are 40 and 21, respectively. If the mean deviation about the median is 26, then $2a$ is equal to:

1. A.109
2. B.117
3. C.161
4. D.131

For 10 observations $x_1,x_2,\dots ,x_{10}$, if ${\sum }_{i=1}^{10}(x_i+2{)}^2=180$ and ${\sum }_{i=1}^{10}(x_i-1{)}^2=90$, then their standard deviation is:

1. A.2
2. B.$\sqrt{3}$
3. C.$2\sqrt{2}$
4. D.3

If the mean of the data $$\begin{array}{lcccccc}\text{Class}& 5\text{-}10& 10\text{-}15& 15\text{-}20& 20\text{-}25& 25\text{-}30& 30\text{-}35\\ \text{Frequency}& 2& k& 28& 54& k+1& 5\end{array}$$ is $21$, then $\mathrm{k}$ is one of the roots of the equation :

1. A.$2x^2-23x-10=0$
2. B.$4x^2-35x+24=0$
3. C.$2x^2-19x-10=0$
4. D.$2x^2-35x+98=0$

Given below are two statements : Statement I: The second ionization enthalpy of B, Al and Ga is in the order of B > Al > Ga. Statement II: The correct order in terms of first ionization enthalpy is Si < Ge < Pb < Sn. In the light of the above statements, choose the correct answer from the options given below :

1. A.Both Statement I and Statement II are true
2. B.Both Statement I and Statement II are false
3. C.Statement I is true but Statement II is false
4. D.Statement I is false but Statement II is true

The mean and variance of $10$ observations are $9$ and $34.2$ , respectively. If $8$ of these observations are $2,3,5,10,11,13,15,21$, then the mean deviation about the median of all the $10$ observations is

1. A.7
2. B.4
3. C.5
4. D.6

The mean and variance of a data of 10 observations are 10 and 2 , respectively. If an observations $\alpha$ in this data is replaced by $\beta$, then the mean and variance become 10.1 and 1.99 , respectively. Then $\alpha +\beta$ equals

1. A.15
2. B.10
3. C.5
4. D.20

Let the mean and variance of 8 numbers $-10,-7,-1,x,y,9,2,16$ be $\frac{7}{2}$ and $\frac{293}{4}$, respectively. Then the mean of 4 numbers $x,y,x+y+1,|x-y|$ is :

1. A.11
2. B.9
3. C.10
4. D.12

The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is ____ .

If the mean and the variance of the data $$\begin{array}{ccccc}\text{Class}& 4-8& 8-12& 12-16& 16-20\\ \text{Frequency}& 3& \lambda & 4& 7\end{array}$$ are $\mu$ and 19 respectively, then the value of $\lambda +\mu$ is :

1. A.21
2. B.19
3. C.18
4. D.20

Let the mean and variance of 7 observations $2,4,10,x,12,14,y,x>y$, be 8 and 16 respectively. Two numbers are chosen from $\{1,2,3,x-4,y,5\}$ one after another without replacement, then the probability, that the smaller number among the two chosen numbers is less than 4 , is :

1. A.$\frac{4}{5}$
2. B.$\frac{3}{5}$
3. C.$\frac{2}{5}$
4. D.$\frac{1}{3}$

A random variable X takes values 0, 1, 2, 3 with probabilities $\frac{2a+1}{30},\frac{8a-1}{30},\frac{4a+1}{30},b$ respectively, where $a,b\in \mathbb{R}$. Let $\mu$ and $\sigma$ respectively be the mean and standard deviation of $X$ such that ${\sigma }^2+{\mu }^2=2$. Then $\frac{a}{b}$ is equal to:

1. A.12
2. B.60
3. C.30
4. D.3

A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let $X$ denote the number of defective pens. Then the variance of $X$ is

1. A.$\frac{11}{15}$
2. B.$\frac{2}{15}$
3. C.$\frac{3}{5}$
4. D.$\frac{28}{75}$

Let the mean and the standard deviation of the observation $2,3,3,4,5,7,a,b$ be 4 and $\sqrt{2}$ respectively. Then the mean deviation about the mode of these observations is :

1. A.$\frac{1}{2}$
2. B.$\frac{3}{4}$
3. C.1
4. D.2

Let the Mean and Variance of five observations $x_1=1,x_2=3,x_3=a,x_4=7$ and $x_5=b,a>b$, be $5$ and $10$ respectively. Then the Variance of the observations $n+x_n,n=1,2,\dots ,5$ is

1. A.$17$
2. B.$16$
3. C.$16.4$
4. D.$17.4$

If the mean and the variance of $6,4,a,8,b,12,10,13$ are 9 and 9.25 respectively, then $a+b+ab$ is equal to :

1. A.103
2. B.106
3. C.100
4. D.105

Three defective oranges are accidentally mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If $x$ denotes the number of defective oranges, then the variance of $x$ is

1. A.$26/75$
2. B.$14/25$
3. C.$18/25$
4. D.$28/75$

For a statistical data $x_1,x_2,\dots ,x_{10}$ of $10$ values, a student obtained the mean as $5.5$ and ${\sum }_{i=1}^{10}{x}_i^2=371$. He later found that he had noted two values in the data incorrectly as $4$ and $5$ , instead of the correct values $6$ and $8$ , respectively. The variance of the corrected data is

1. A.$5$
2. B.$7$
3. C.$9$
4. D.$4$

The variance of the numbers $8,21,34,47,\dots ,320$ is _________.

A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$ and ${\sigma }^2$ denote the mean and variance of $X$, then the value of $64\left(\mu +{\sigma }^2\right)$ is:

1. A.64
2. B.32
3. C.51
4. D.48

The frequency distribution of the age of students in a class of 40 students is given below. | Age | 15 | 16 | 17 | 18 | 19 | 20 | |---|---|---|---|---|---|---| | No of Students | 5 | 8 | 5 | 12 | $x$ | $y$ | If the mean deviation about the median is 1.25, then $4x+5y$ is equal to :

1. A.43
2. B.46
3. C.44
4. D.47