MHT CET · Maths · Statistics
The mean and the standard deviation of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by p and then reduced by \(q\), where \(p \neq 0\) and \(q \neq 0\). If the new mean and new standard deviation (s.d.) become half of the original values, then q is equal to
- A \(\quad-20\)
- B -5
- C 10
- D -10
Answer & Solution
Correct Answer
(A) \(\quad-20\)
Step-by-step Solution
Detailed explanation
Mean \(=20\) and \(\mathrm{SD}=2, \mathrm{n}=10\)
If each observation is multiplied by p and then reduced by q.
New Mean \(=\bar{x}_1=\mathrm{p} \bar{x}-\mathrm{q}\)
\(\begin{aligned}
& 10=p(20)-q \\
& \Rightarrow 20 p-q=10
...(i)\end{aligned}\)
New \(\mathrm{SD}=\sigma_1=|\mathrm{p}| \sigma\)
Squaring on both sides,
\(\begin{aligned}
& \Rightarrow 1=\mathrm{p}^2 \times 4 \\
& \Rightarrow \mathrm{p}^2=\frac{1}{4} \\
& \Rightarrow \mathrm{p}= \pm \frac{1}{2}
\end{aligned}\)
From (i)
When, \(\mathrm{p}=\frac{-1}{2}, \mathrm{q}=-20\)
When, \(\mathrm{p}=\frac{1}{2}, \mathrm{q}=0\)
If each observation is multiplied by p and then reduced by q.
New Mean \(=\bar{x}_1=\mathrm{p} \bar{x}-\mathrm{q}\)
\(\begin{aligned}
& 10=p(20)-q \\
& \Rightarrow 20 p-q=10
...(i)\end{aligned}\)
New \(\mathrm{SD}=\sigma_1=|\mathrm{p}| \sigma\)
Squaring on both sides,
\(\begin{aligned}
& \Rightarrow 1=\mathrm{p}^2 \times 4 \\
& \Rightarrow \mathrm{p}^2=\frac{1}{4} \\
& \Rightarrow \mathrm{p}= \pm \frac{1}{2}
\end{aligned}\)
From (i)
When, \(\mathrm{p}=\frac{-1}{2}, \mathrm{q}=-20\)
When, \(\mathrm{p}=\frac{1}{2}, \mathrm{q}=0\)
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