Statistics
Statistics
nta_abhyas_2025
Grade 11
Question:
Let $x_1, x_2, x_3, \ldots, x_k$ be $k$ observations and $w_i = ax_i + b$ for $i = 1, 2, 3, \ldots, k$, where $a$ and $b$ are constants. If mean of $x_i$ is 52 and their standard deviation is 12 and mean of $w_i$ is 60 and their standard deviation is 15, then the value of $a$ and $b$ should be
$a = 1.25, b = -5$
$a = 1.25, b = 5$
$a = 2.5, b = -5$
$a = 2.5, b = 5$
Step-by-Step Solution
Key Concept: The standard deviation scales linearly with the coefficient $a$, and solving simultaneous equations gives the regression parameters.
Given $u_0 = az_r + b$ where $u_0 = 60$, $a = 52$, we have $60 = a(52) + b$. From the standard deviation relationship, $\text{S.D of } u = |a|(\text{S.D of } x)$, so $15 = |a| \cdot 12$, giving $a = \pm 1.25$. Solving with both values of $a$: if $a = 1.25$, then $60 = 65 + b$, so $b = -5$; if $a = -1.25$, then $60 = -65 + b$, so $b = 125$. The value $b = -5$ is the required answer.
Correct Answer: 5