Let X={mathrm{x} ∈ mathrm{N}: 1 ≤ mathrm{x} ≤ 17} and mathrm{Y}={mathrm{ax}+mathrm{b}: mathrm{

English

Gujarati

Hindi

13.Statistics

hard

Let $X=\{\mathrm{x} \in \mathrm{N}: 1 \leq \mathrm{x} \leq 17\}$ and $\mathrm{Y}=\{\mathrm{ax}+\mathrm{b}: \mathrm{x} \in \mathrm{X}$ and $\mathrm{a}, \mathrm{b} \in \mathrm{R}, \mathrm{a}>0\} .$ If mean and variance of elements of $Y$ are $17$ and $216$ respectively then $a + b$ is equal to

$-7$

$7$

$9$

$-27$

(JEE MAIN-2020)

Solution

$\sigma^{2}=$ variance

$\mu=$ mean

$\sigma^{2}=\frac{\sum_{i=1}^{n}\left(x_{i}-\mu\right)^{2}}{n}$

$\mu=17$

$\Rightarrow \frac{\sum_{x=1}^{17}(a x+b)}{17}=17$

$\Rightarrow \quad 9 a+b=17$

$\sigma^{2}=216$

$\Rightarrow \quad \frac{\sum_{x=1}^{17}(a x+b-17)^{2}}{17}=216$

$\Rightarrow \frac{\sum_{x=1}^{17} a^{2}(x-9)^{2}}{17}=216$

$\Rightarrow \quad a^{2} 81-18 \times 9 a^{2}+a^{2} 3 \times(35)=216$

$\Rightarrow \quad$ From $(1), b=-10$

So, $a+b=-7$

Standard 11

Mathematics

If each observation of a raw data whose variance is ${\sigma ^2}$, is multiplied by $\lambda$, then the variance of the new set is

normal

Find the mean, variance and standard deviation using short-cut method

Height in cms $70-75$ $75-80$ $80-85$ $85-90$ $90-95$ $95-100$ $100-105$ $105-110$ $110-115$

No. of children $3$ $4$ $7$ $7$ $15$ $9$ $6$ $6$ $3$

hard

Find the mean and variance for the first $10$ multiples of $3$

medium

If mean and standard deviation of $5$ observations $x_1 ,x_2 ,x_3 ,x_4 ,x_5$ are $10$ and $3$, respectively, then the variance of $6$ observations $x_1 ,x_2 ,…..,x_3$ and $-50$ is equal to

hard

(JEE MAIN-2019)

What is the standard deviation of the following series

class $0-10$ $10-20$ $20-30$ $30-40$

Freq $1$ $3$ $4$ $2$

medium

Height in cms	$70-75$	$75-80$	$80-85$	$85-90$	$90-95$	$95-100$	$100-105$	$105-110$	$110-115$
No. of children	$3$	$4$	$7$	$7$	$15$	$9$	$6$	$6$	$3$

Let $X=\{\mathrm{x} \in \mathrm{N}: 1 \leq \mathrm{x} \leq 17\}$ and $\mathrm{Y}=\{\mathrm{ax}+\mathrm{b}: \mathrm{x} \in \mathrm{X}$ and $\mathrm{a}, \mathrm{b} \in \mathrm{R}, \mathrm{a}>0\} .$ If mean and variance of elements of $Y$ are $17$ and $216$ respectively then $a + b$ is equal to

Solution

Similar Questions

If each observation of a raw data whose variance is ${\sigma ^2}$, is multiplied by $\lambda$, then the variance of the new set is

Find the mean, variance and standard deviation using short-cut method Height in cms $70-75$ $75-80$ $80-85$ $85-90$ $90-95$ $95-100$ $100-105$ $105-110$ $110-115$ No. of children $3$ $4$ $7$ $7$ $15$ $9$ $6$ $6$ $3$

Find the mean and variance for the first $10$ multiples of $3$

If mean and standard deviation of $5$ observations $x_1 ,x_2 ,x_3 ,x_4 ,x_5$ are $10$ and $3$, respectively, then the variance of $6$ observations $x_1 ,x_2 ,…..,x_3$ and $-50$ is equal to

What is the standard deviation of the following series class $0-10$ $10-20$ $20-30$ $30-40$ Freq $1$ $3$ $4$ $2$

Start a Free Trial Now

Find the mean, variance and standard deviation using short-cut method

Height in cms $70-75$ $75-80$ $80-85$ $85-90$ $90-95$ $95-100$ $100-105$ $105-110$ $110-115$

No. of children $3$ $4$ $7$ $7$ $15$ $9$ $6$ $6$ $3$

What is the standard deviation of the following series

class $0-10$ $10-20$ $20-30$ $30-40$

Freq $1$ $3$ $4$ $2$