The mean of $n$ observations is $\bar{x}$. If three observations $n+1, n-1, 2n-1$ are added such that the mean remains the same,then the value of $n$ is

Let $\bar{x}, M$ and $\sigma^2$ be respectively the mean,mode and variance of $n$ observations $x_1, x_2, ..., x_n$ and $d_i = -x_i - a, i = 1, 2, ..., n$,where $a$ is any number. Statement $I$: Variance of $d_1, d_2, ..., d_n$ is $\sigma^2$. Statement $II$: Mean and mode of $d_1, d_2, ..., d_n$ are $-\bar{x} - a$ and $-M - a$,respectively.

In a town, the probability that a sick person may need to be admitted to an $ICU$ is $10 \%$. If the probability that a person getting admitted to an $ICU$ goes above $5 \%$, then the threat level is raised. The minimum percentage of the population of the town that should fall sick in order to raise the threat level is

For a group of $100$ students,the mean $\bar{x}_1$ and the standard deviation $\sigma_1$ of their marks were found to be $40$ and $15$ respectively. Later,it was observed that the scores $40$ and $50$ were misread as $30$ and $60$ respectively. If the mean and the standard deviation with the corrected observations of the scores are $\bar{x}_2$ and $\sigma_2$ respectively,then:

The mean and variance of $20$ observations are found to be $10$ and $4,$ respectively. On rechecking,it was found that an observation $9$ was incorrect and the correct observation was $11$. Then the correct variance is

The average marks of boys in a class is $40$ and that of girls is $45$. The average marks of both boys and girls combined is $42$. Then the percentage of boys in the class is (in $\%$)