Question

Let $x_1,x_2,...,x_n$ be n observations . Let $y_1=ax+b$ for i = 1, 2 ,...., n , where a and b are standard deviation of y1 is 15, the values of a and b are

• A. a = 1.25, b = -5
• B. a = 1.25, b = 5
• C. a = 2.5, b= -5
• D. a =2.5, b= 5

Answer 1

The correct option is A

a = 1.25, b = -5

It is given that $y_i=a{x}_i+b$ for i =1,2, 3, ....,n, where a and b are constants.

$\overline{x_i}=48$ and ${\sigma }_{xi}=12$

$\overline{y_i}=55$ and ${\sigma }_{yi}=15$

$y_i=a{x}_i+b$

$\Rightarrow \frac{\sum y_i}{n}=\frac{\sum (a{x}_i+b)}{n}$ $\Rightarrow \frac{\sum y_i}{n}=a\frac{\sum x_i}{n}+\frac{\sum b}{n}\Rightarrow \overline{y_i}=a\overline{x_i}+b$

$\Rightarrow 55=48a+b$ ..... (1)

Now,

Standard deviation of $y_i$ = Standard deviation of $a{x}_i+b$

$\Rightarrow {\sigma }_{yi}=\sigma \times {\sigma }_{xi}\Rightarrow 15=12a\Rightarrow a=\frac{15}{12}=1.25$

Putting a = 1.25 in (1), we get

$b=55-48\times 1.25=55-60=-5$

Thus, the values of a and b are 1.25 and -5 , respectively.

Hence, the correct answer is option (a).