Test Index

Statistics Part 1

Section: Mathematics

© examsnet.com

Question : 38 of 100

Marks: +1, -0

Let in a series of

2 n

observations, half of them are equal to

a

and remaining half are equal to

-a

. Also, by adding a constant

b

in each of these observations, the mean and standard deviation of new set become 5 and 20 , respectively. Then, the value of

a^{2}+b^{2}

[18 Mar 2021 Shift 2]

425
650
250
925

Solution:

Let observations are denoted by

x_{i}

for

1 \leq i < 2 n

.

\therefore \text{Mean}(\overline{x})=\frac{\sum\limits_{x_{i}}}{2 n}

= \frac{(a+a+a+\dots +a)-(a+a+a+\dots +a)}{2 n}

\Rightarrow \qquad \overline{x}=0

and

\sigma_{x}^{2} = \frac{\sum x_{1}^{2}}{2 n} - (\overline{x})^{2} = \frac{\overbrace{a^{2}+a^{2}+\dots +a^{2}}^{2}}{2 n} - 0 = a^{2}

\Rightarrow \qquad \sigma_{x}^{2}=a^{2} \Rightarrow \sigma_{x}=a

According to the question, adding a constant

b

, then new mean

(\overline{y})=\overline{x}+b=5

\Rightarrow \qquad 0+b=5

\Rightarrow \qquad b=5

and new SD

(\sigma_{y})=\sigma_{x}

(no change in SD)

\Rightarrow \; \sigma_{y}=\sigma_{x}=20

\Rightarrow \; a=20

\; \therefore a^{2}+b^{2}=400+25=425

© examsnet.com

Go to Question:

Prev Question

Next Question