Statistics · JEE Main previous year questions

Let the mean and variance of $12$ observations be $\frac{9}{2}$ and $4$ respectively. Later on, it was observed that two observations were considered as $9$ and $10$ instead of $7$ and $14$ respectively. If the correct variance is $\frac{\mathrm{m}}{\mathrm{n}}$, where $\mathrm{m}$ and $\mathrm{n}$ are coprime, then $\mathrm{m}+\mathrm{n}$ is equal to :

1. A.$317$
2. B.$316$
3. C.$314$
4. D.$315$

The mean and variance of a set of 15 numbers are 12 and 14 respectively. The mean and variance of another set of 15 numbers are 14 and ${\sigma }^2$ respectively. If the variance of all the 30 numbers in the two sets is 13 , then ${\sigma }^2$ is equal to :

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

If the mean and variance of the frequency distribution | $x_i$ | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $f_i$ | 4 | 4 | $\alpha$ | 15 | 8 | $\beta$ | 4 | 5 | are 9 and 15.08 respectively, then the value of ${\alpha }^2+{\beta }^2-\alpha \beta$ is ________.

The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1, 3, 5, then the sum of cubes of the remaining two observations is :

1. A.1792
2. B.1216
3. C.1456
4. D.1072

In a binomial distribution $B(n,p)$, the sum and the product of the mean and the variance are 5 and 6 respectively, then $6(n+p-q)$ is equal to :

1. A.$52$
2. B.$50$
3. C.$51$
4. D.$53$

If the variance of the frequency distribution | $x_i$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Frequency $f_i$ | 3 | 6 | 16 | $\alpha$ | 9 | 5 | 6 | is 3, then $\alpha$ is equal to ___________.

Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and $\alpha (>0)$, and the mean and standard deviation of marks of class $B$ of $n$ students be respectively 55 and $30-\alpha$. If the mean and variance of the marks of the combined class of $100+n$ students are respectively 50 and 350 , then the sum of variances of classes $A$ and $B$ is :

1. A.450
2. B.900
3. C.650
4. D.500

The mean and variance of 7 observations are 8 and 16 respectively. If one observation 14 is omitted and a and b are respectively mean and variance of remaining 6 observations, then $a+3b-5$ is equal to ____.

Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $\mu$ and ${\sigma }^2$ represent mean and variance of X, respectively, then $10({\mu }^2+{\sigma }^2)$ is equal to :

1. A.20
2. B.30
3. C.250
4. D.25

Let $X=\{11,12,13,\dots ,40,41\}$ and $Y=\{61,62,63,\dots ,90,91\}$ be the two sets of observations. If $\bar{x}$ and $\bar{y}$ are their respective means and ${\sigma }^2$ is the variance of all the observations in $\mathrm{X}\cup \mathrm{Y}$, then $\left|\bar{x}+\bar{y}-{\sigma }^2\right|$ is equal to ____.

The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2, then their new variance is equal to :

1. A.3.92
2. B.4.08
3. C.3.96
4. D.4.04

Let the six numbers ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,{\mathrm{a}}_4,{\mathrm{a}}_5,{\mathrm{a}}_6$, be in A.P. and ${\mathrm{a}}_1+{\mathrm{a}}_3=10$. If the mean of these six numbers is $\frac{19}{2}$ and their variance is ${\sigma }^2$, then $8{\sigma }^2$ is equal to :

1. A.220
2. B.210
3. C.105
4. D.200

Let the mean and the variance of 20 observations $x_1,x_2,\dots ,x_{20}$ be 15 and 9 , respectively. For $\alpha \in \mathbf{R}$, if the mean of ${\left(x_1+\alpha \right)}^2,{\left(x_2+\alpha \right)}^2,\dots ,{\left(x_{20}+\alpha \right)}^2$ is 178 , then the square of the maximum value of $\alpha$ is equal to ____.

The sum and product of the mean and variance of a binomial distribution are 82.5 and 1350 respectively. Then the number of trials in the binomial distribution is ____________.

The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is _____________.

Let $\mathrm{X}$ have a binomial distribution $\mathrm{B}(n,p)$ such that the sum and the product of the mean and variance of $\mathrm{X}$ are $24$ and $128$ respectively. If $P(X>n-3)=\frac{k}{2^n}$, then $\mathrm{k}$ is equal to :

1. A.$528$
2. B.$529$
3. C.$629$
4. D.$630$

The mean and variance of a binomial distribution are $\alpha$ and $\frac{\alpha }{3}$ respectively. If $\mathrm{P}(X=1)=\frac{4}{243}$, then $\mathrm{P}(X=4\text{ or }5)$ is equal to :

1. A.$\frac{5}{9}$
2. B.$\frac{64}{81}$
3. C.$\frac{16}{27}$
4. D.$\frac{145}{243}$

If the mean deviation about median for the numbers 3, 5, 7, 2k, 12, 16, 21, 24, arranged in the ascending order, is 6 then the median is :

1. A.11.5
2. B.10.5
3. C.12
4. D.11

Suppose a class has 7 students. The average marks of these students in the mathematics examination is 62, and their variance is 20. A student fails in the examination if he/she gets less than 50 marks, then in worst case, the number of students can fail is _________.

Let $X$ be a random variable having binomial distribution $B(7,p)$. If $P(X=3)=5P(X=4)$, then the sum of the mean and the variance of $X$ is :

1. A.$\frac{105}{16}$
2. B.$\frac{7}{16}$
3. C.$\frac{77}{36}$
4. D.$\frac{49}{16}$

The mean and variance of the data $4,5,6,6,7,8,x,y$, where $x<y$, are $6$ and $\frac{9}{4}$ respectively. Then $x^4+y^2$ is equal to :

1. A.$162$
2. B.$320$
3. C.$674$
4. D.$420$

The mean of the numbers a, b, 8, 5, 10 is 6 and their variance is 6.8. If M is the mean deviation of the numbers about the mean, then 25 M is equal to :

1. A.60
2. B.55
3. C.50
4. D.45

The mean and standard deviation of 50 observations are 15 and 2 respectively. It was found that one incorrect observation was taken such that the sum of correct and incorrect observations is 70. If the correct mean is 16, then the correct variance is equal to :

1. A.10
2. B.36
3. C.43
4. D.60

If the mean deviation about the mean of the numbers $1,2,3,\dots ,n$, where n is odd, is $\frac{5(n+1)}{n}$, then n is equal to $\underline{}$.

Let X be a random variable with distribution. | x | -2 | -1 | 3 | 4 | 6 | | :--- | :--- | :--- | :--- | :--- | :--- | | $\mathrm{P}(\mathrm{X}=\mathrm{x})$ | $\frac{1}{5}$ | a | $\frac{1}{3}$ | $\frac{1}{5}$ | b | If the mean of X is 2.3 and variance of X is ${\sigma }^2$, then $100{\sigma }^2$ is equal to :