Statistics · JEE Main previous year questions

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2023 23 questions

Q62023 · 13 Apr 2023 · Shift 1
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 $X$ denotes the number of tosses of the coin, then the mean of $X$ is :
1. A. $\frac{81}{64}$
2. B. $\frac{37}{16}$
3. C. $\frac{21}{16}$
4. D. $\frac{15}{16}$
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Answer: C
Q212023 · 13 Apr 2023 · Shift 1· Numerical answer
Let the mean of the data | $x$ | 1 | 3 | 5 | 7 | 9 | | :--- | :--- | :--- | :--- | :--- | :--- | | Frequency ( $f$ ) | 4 | 24 | 28 | $α$ | 8 | be 5. If $m$ and $σ^{2}$ are respectively the mean deviation about the mean and the variance of the data, then $\frac{3 α}{m + σ ^{2}}$ is equal to ________
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Answer: 8
Q202023 · 13 Apr 2023 · Shift 2
The random variable $X$ follows binomial distribution $B (n, p)$ , for which the difference of the mean and the variance is $1$ . If $2 P (X = 2) = 3 P (X = 1)$ , then $n^{2} P (X > 1)$ is equal to :
1. A. $15$
2. B. $12$
3. C. $11$
4. D. $16$
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Answer: C
Q212023 · 13 Apr 2023 · Shift 2· Numerical answer
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 _________
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Answer: 269
Q12023 · 12 Apr 2023 · Shift 1
Two dice A and B are rolled. Let the numbers obtained on A and B be $α$ and $β$ respectively. If the variance of $α - β$ 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. $36$
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Answer: A
Q282023 · 12 Apr 2023 · Shift 1· Numerical answer
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 _________.
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Answer: 211
Q72023 · 11 Apr 2023 · Shift 1
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
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Answer: C
Q92023 · 11 Apr 2023 · Shift 2
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. $3$
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Answer: B
Q242023 · 10 Apr 2023 · Shift 1· Numerical answer
If the mean of the frequency distribution $Class : Frequency : 0 - 10 2 10 - 20 3 20 - 30 x 30 - 40 5 40 - 50 4$ is 28, then its variance is _________.
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Answer: 151
Q82023 · 10 Apr 2023 · Shift 2
Let $μ$ be the mean and $σ$ be the standard deviation of the distribution $x_{i} f_{i} 0 k + 2 1 2 k 2 k^{2} - 1 3 k^{2} - 1 4 k^{2} + 1 5 k - 3$ where $\sum f_{i} = 62$ . If $[x]$ denotes the greatest integer $\leq x$ , then $[μ^{2} + σ^{2}]$ is equal to :
1. A.9
2. B.8
3. C.6
4. D.7
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Answer: B
Q242023 · 8 Apr 2023 · Shift 1· Numerical answer
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 $3 x - 2 y$ is equal to ___________.
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Answer: 25
Q62023 · 8 Apr 2023 · Shift 2
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 $\frac{m}{n}$ , where $m$ and $n$ are coprime, then $m + n$ is equal to :
1. A. $317$
2. B. $316$
3. C. $314$
4. D. $315$
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Answer: A
Q52023 · 6 Apr 2023 · Shift 1
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 $σ^{2}$ respectively. If the variance of all the 30 numbers in the two sets is 13 , then $σ^{2}$ is equal to :
1. A.12
2. B.11
3. C.10
4. D.9
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Answer: C
Q212023 · 6 Apr 2023 · Shift 2· Numerical answer
If the mean and variance of the frequency distribution | $x_{i}$ | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $f_{i}$ | 4 | 4 | $α$ | 15 | 8 | $β$ | 4 | 5 | are 9 and 15.08 respectively, then the value of $α^{2} + β^{2} - α β$ is ________.
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Answer: 25
Q82023 · 1 Feb 2023 · Shift 1
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
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Answer: D
Q172023 · 1 Feb 2023 · Shift 1
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. $53$
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Answer: A
Q282023 · 31 Jan 2023 · Shift 1· Numerical answer
If the variance of the frequency distribution | $x_{i}$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Frequency $f_{i}$ | 3 | 6 | 16 | $α$ | 9 | 5 | 6 | is 3, then $α$ is equal to ___________.
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Answer: 5
Q62023 · 31 Jan 2023 · Shift 2
Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and $α (> 0)$ , and the mean and standard deviation of marks of class $B$ of $n$ students be respectively 55 and $30 - α$ . 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
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Answer: D
Q272023 · 30 Jan 2023 · Shift 1· Numerical answer
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 $a + 3 b - 5$ is equal to ____.
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Answer: 37
Q32023 · 29 Jan 2023 · Shift 1
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 $μ$ and $σ^{2}$ represent mean and variance of X, respectively, then $10 (μ^{2} + σ^{2})$ is equal to :
1. A.20
2. B.30
3. C.250
4. D.25
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Answer: A
Q222023 · 29 Jan 2023 · Shift 2· Numerical answer
Let $X = {11, 12, 13, \dots, 40, 41}$ and $Y = {61, 62, 63, \dots, 90, 91}$ be the two sets of observations. If $\overset{x}{ˉ}$ and $\overset{y}{ˉ}$ are their respective means and $σ^{2}$ is the variance of all the observations in $X \cup Y$ , then $\overset{x}{ˉ} + \overset{y}{ˉ} - σ^{2}$ is equal to ____.
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Answer: 603
Q42023 · 25 Jan 2023 · Shift 1
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
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Answer: C
Q112023 · 24 Jan 2023 · Shift 2
Let the six numbers $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ , be in A.P. and $a_{1} + a_{3} = 10$ . If the mean of these six numbers is $\frac{19}{2}$ and their variance is $σ^{2}$ , then $8 σ^{2}$ is equal to :
1. A.220
2. B.210
3. C.105
4. D.200
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Answer: B