Let $x_1, x_2, \ldots, x_{11}$ be the observations satisfying $\sum_{i=1}^{11}(x_i-4)=22$ and $\sum_{i=1}^{11}(x_i-4)^2=154$. If the mean and variance of the observations are $\alpha$ and $\beta$,then the quadratic equation having the roots $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ is

There are $n$ observations and all of them are negative numbers. The ascending order of these observations is $x_1, x_2, \ldots, x_n$. If the signs of the first term and last term in that order are changed,then the range of the data is

If the mean of the data:

Class	$5 - 10$	$10 - 15$	$15 - 20$	$20 - 25$	$25 - 30$	$30 - 35$
Frequency	$2$	$k$	$28$	$54$	$k + 1$	$5$

is $21$, then $k$ is one of the roots of the equation:

Let $n \geq 3$. $A$ list of numbers $x_1, x_2, \ldots, x_n$ has mean $\mu$ and standard deviation $\sigma$. $A$ new list of numbers $y_1, y_2, \ldots, y_n$ is made as follows: $y_1 = \frac{x_1+x_2}{2}$,$y_2 = \frac{x_1+x_2}{2}$ and $y_j = x_j$ for $j = 3, 4, \ldots, n$. The mean and the standard deviation of the new list are $\hat{\mu}$ and $\hat{\sigma}$. Which of the following is necessarily true?

The range of the observations $2, 3, 5, 9, 8, 7, 6, 5, 7, 4, 3$ is . . . . . . .

The mean and variance of $8$ observations are $10$ and $13.5,$ respectively. If $6$ of these observations are $5, 7, 10, 12, 14, 15,$ then the absolute difference of the remaining two observations is