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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:

Let \( \mu \) be the mean and \( \sigma \) be the standard deviation of the distribution,

Where \( \sum f_{i} = 62 \). If \( [x] \) denotes the greatest integer \( \le x \), then \( \left[ \mu^{2} + \sigma^{2} \right] \) 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 ________ .

Let \( 9 = x_1 < x_2 < \ldots < x_7 \) be in an A.P. with common difference \( d \). If the standard deviation of \( x_1, x_2, \ldots, x_7 \) is \( 4 \) and the mean is \( \bar{x} \), then \( \bar{x} + x_6 \) 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:

If the mean and variance of the frequency distribution

are \( 9 \) and \( 15.08 \) respectively, then the value of \( \alpha^{2}+\beta^{2}-\alpha\beta \) is __________.

Let the mean of the data

Be \( 5 \). If \( m \) and \( \sigma^2 \) are respectively the mean deviation about the mean and the variance of the data, then \( \dfrac{3\alpha}{m + \sigma^2} \) is equal to _____.

Let \( S \) be the set of all values of \( a_{1} \) for which the mean deviation about the mean of \( 100 \) consecutive positive integers \( a_{1}, a_{2}, a_{3}, \ldots, a_{100} \) is \( 25 \). Then \( S \) is:

Let \( X = \{11, 12, 13, \ldots, 40, 41\} \) and \( Y = \{61, 62, 63, \ldots, 90, 91\} \) 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 \( X \cup Y \), then \( \left| \bar{x} + \bar{y} - \sigma^2 \right| \) is equal to ______.

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 \( 6 \) observation, then \( a+3b-5 \) is equal to \( \_\_\_\_\_ \).

The mean and standard deviation of \( 10 \) observations are \( 20 \) and \( 8 \) respectively. Later on, it was observed that one observation was recorded as \( 50 \) instead of \( 40 \). Then the correct variance is:

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 ________.

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:

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 \( \sigma^2 \), then \( 8\sigma^2 \) is equal to:

If the variance of the frequency distribution

is \( 3 \), then \( \alpha \) is equal to _____.

The mean and variance of \( 5 \) observations are \( 5 \) and \( 8 \) respectively. If \( 3 \) observations are \( 1,3,5 \), then the sum of the cubes of the remaining two observations 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 \( \dfrac{p}{q} \), where \( p \) and \( q \) are co-prime, then the sum of the positive divisors of \( p \) is equal to:

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 __________.

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

Let the mean and variance of \( 12 \) observations be \( \dfrac{9}{2} \) 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 \( \dfrac{m}{n} \), where \( m \) and \( n \) are coprime, then \( m+n \) is equal to:

If the mean of the frequency distribution

is \( 28 \), then its variance is __________.

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:

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JEE Main 2023 31st Jan Morning Shift

If the variance of the frequency distribution

is \( 3 \), then \( \alpha \) is equal to _____.