Statistics · JEE Main previous year questions

If the variance of the frequency distribution $$\begin{array}{ccccccc}x& c& 2c& 3c& 4c& 5c& 6c\\ f& 2& 1& 1& 1& 1& 1\end{array}$$ is $160$, then the value of $c\in N$ is

1. A.$5$
2. B.$8$
3. C.$6$
4. D.$7$

Let $a,b,c\in \mathbf{N}$ and $a<b<c$. Let the mean, the mean deviation about the mean and the variance of the 5 observations $9,25,a,b,c$ be $18,4$ and $\frac{136}{5}$, respectively. Then $2a+b-c$ is equal to ____

Let the mean and the standard deviation of the probability distribution \begin{tabular}{|l|l|l|l|l|} \hline X & $\alpha$ & 1 & 0 & $-3$ \\ \hline P(X) & $\frac{1}{3}$ & K & $\frac{1}{6}$ & $\frac{1}{4}$ \\ \hline \end{tabular} be $\mu$ and $\sigma$, respectively. If $\sigma -\mu =2$, then $\sigma +\mu$ is equal to _________.

Let $\alpha ,\beta \in \mathbf{R}$. Let the mean and the variance of 6 observations $-3,4,7,-6,\alpha ,\beta$ be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is :

1. A.$\frac{16}{3}$
2. B.$\frac{11}{3}$
3. C.$\frac{14}{3}$
4. D.$\frac{13}{3}$

If the mean of the following probability distribution of a random variable $\mathrm{X}$ : $$\begin{array}{llllll}\mathrm{X}& 0& 2& 4& 6& 8\\ \mathrm{P}(\mathrm{X})& a& 2a& a+b& 2b& 3b\end{array}$$ is $\frac{46}{9}$, then the variance of the distribution is

1. A.$\frac{581}{81}$
2. B.$\frac{566}{81}$
3. C.$\frac{151}{27}$
4. D.$\frac{173}{27}$

Let the median and the mean deviation about the median of 7 observations $170,125,230,190,210$, a, b be 170 and $\frac{205}{7}$ respectively. Then the mean deviation about the mean of these 7 observations is :

1. A.31
2. B.28
3. C.30
4. D.32

Consider 10 observations $x_1,x_2,\dots ,x_{10}$ such that ${\sum }_{i=1}^{10}(x_i-\alpha )=2$ and ${\sum }_{i=1}^{10}(x_i-\beta {)}^2=40$, where $\alpha ,\beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and $\frac{84}{25}$ respectively. Then $\frac{\beta }{\alpha }$ is equal to :

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

Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable $x$ to be the number of rotten apples in a draw of two apples, the variance of $x$ is

1. A.$\frac{57}{153}$
2. B.$\frac{40}{153}$
3. C.$\frac{37}{153}$
4. D.$\frac{47}{153}$

Let the mean and the variance of 6 observations $a,b,68,44,48,60$ be $55$ and $194$, respectively. If $a>b$, then $a+3b$ is

1. A.180
2. B.210
3. C.190
4. D.200

Let M denote the median of the following frequency distribution | Class | 0 - 4 | 4 - 8 | 8 - 12 | 12 - 16 | 16 - 20 | |---|---|---|---|---|---| | Frequency | 3 | 9 | 10 | 8 | 6 | Then 20M is equal to :

1. A.104
2. B.52
3. C.208
4. D.416

If the mean and variance of the data $65,68,58,44,48,45,60,\alpha ,\beta ,60$ where $\alpha >\beta$, are 56 and 66.2 respectively, then ${\alpha }^2+{\beta }^2$ is equal to _______.

If the mean and variance of five observations are $\frac{24}{5}$ and $\frac{194}{25}$ respectively and the mean of the first four observations is $\frac{7}{2}$, then the variance of the first four observations is equal to

1. A.$\frac{5}{4}$
2. B.$\frac{4}{5}$
3. C.$\frac{105}{4}$
4. D.$\frac{77}{12}$

Let ${\mathrm{a}}_1,{\mathrm{a}}_2,\dots {\mathrm{a}}_{10}$ be 10 observations such that $\sum _{\mathrm{k}=1}^{10}{\mathrm{a}}_{\mathrm{k}}=50$ and $\sum _{\forall \mathrm{k}<\mathrm{j}}{\mathrm{a}}_{\mathrm{k}}\cdot {\mathrm{a}}_{\mathrm{j}}=1100$. Then the standard deviation of ${\mathrm{a}}_1,{\mathrm{a}}_2,\dots ,{\mathrm{a}}_{10}$ is equal to :

1. A.5
2. B.$\sqrt{115}$
3. C.10
4. D.$\sqrt{5}$

The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking it was found that an observation was read as 10 in place of 12 . If $\mu$ and ${\sigma }^2$ denote the mean and variance of the correct observations respectively, then $15\left(\mu +{\mu }^2+{\sigma }^2\right)$ is equal to _______.

A coin is biased so that the head is $3$ times as likely to occur as tail. This coin is tossed until a head or three tails occur. If $\mathrm{X}$ denotes the number of tosses of the coin, then the mean of $\mathrm{X}$ is :

1. A.$\frac{81}{64}$
2. B.$\frac{37}{16}$
3. C.$\frac{21}{16}$
4. D.$\frac{15}{16}$

Let the mean of the data | $x$ | 1 | 3 | 5 | 7 | 9 | | :--- | :--- | :--- | :--- | :--- | :--- | | Frequency ($f$) | 4 | 24 | 28 | $\alpha$ | 8 | be 5. If $m$ and ${\sigma }^2$ are respectively the mean deviation about the mean and the variance of the data, then $\frac{3\alpha }{m+{\sigma }^2}$ is equal to ________

The random variable $\mathrm{X}$ follows binomial distribution $\mathrm{B}(n,p)$, for which the difference of the mean and the variance is $1$ . If $2\mathrm{P}(\mathrm{X}=2)=3\mathrm{P}(\mathrm{X}=1)$, then $n^2\mathrm{P}(\mathrm{X}>1)$ is equal to :

1. A.$15$
2. B.$12$
3. C.$11$
4. D.$16$

The mean and standard deviation of the marks of 10 students were found to be 50 and 12 respectively. Later, it was observed that two marks 20 and 25 were wrongly read as 45 and 50 respectively. Then the correct variance is _________

Two dice A and B are rolled. Let the numbers obtained on A and B be $\alpha$ and $\beta$ respectively. If the variance of $\alpha -\beta$ is $\frac{p}{q}$, where $p$ and $q$ are co-prime, then the sum of the positive divisors of $p$ is equal to :

1. A.$48$
2. B.$31$
3. C.$72$
4. D.$36$

Let the positive numbers $a_1,a_2,a_3,a_4$ and $a_5$ be in a G.P. Let their mean and variance be $\frac{31}{10}$ and $\frac{m}{n}$ respectively, where $m$ and $n$ are co-prime. If the mean of their reciprocals is $\frac{31}{40}$ and $a_3+a_4+a_5=14$, then $m+n$ is equal to _________.

Let sets A and B have 5 elements each. Let the mean of the elements in sets A and B be 5 and 8 respectively and the variance of the elements in sets A and B be 12 and 20 respectively. A new set C of 10 elements is formed by subtracting 3 from each element of A and adding 2 to each element of B. Then the sum of the mean and variance of the elements of C is _________.

1. A.36
2. B.40
3. C.38
4. D.32

Let the mean of 6 observations $1,2,4,5,x$ and $y$ be $5$ and their variance be $10$. Then their mean deviation about the mean is equal to :

1. A.$\frac{10}{3}$
2. B.$\frac{8}{3}$
3. C.$\frac{7}{3}$
4. D.$3$

If the mean of the frequency distribution $$\begin{array}{lccccc}\text{Class :}& 0\text{-}10& 10\text{-}20& 20\text{-}30& 30\text{-}40& 40\text{-}50\\ \text{Frequency :}& 2& 3& x& 5& 4\end{array}$$ is 28, then its variance is _________.

Let $\mu$ be the mean and $\sigma$ be the standard deviation of the distribution $$\begin{array}{ccccccc}{x}_i& 0& 1& 2& 3& 4& 5\\ f_i& k+2& 2k& k^2-1& k^2-1& k^2+1& k-3\end{array}$$ where $\sum f_i=62$. If $[x]$ denotes the greatest integer $\le x$, then $[{\mu }^2+{\sigma }^2]$ is equal to :

1. A.9
2. B.8
3. C.6
4. D.7

Let the mean and variance of $8$ numbers $x,y,10,12,6,12,4,8$ be $9$ and $9.25$ respectively. If $x>y$, then $3x-2y$ is equal to ___________.