A) \[\left( \frac{a}{c} \right)\,\sigma \]
B) \[\left| \frac{a}{c} \right|\,\sigma \]
C) \[\left( \frac{{{a}^{2}}}{{{c}^{2}}} \right)\,\sigma \]
D) None of these
Correct Answer: B
Solution :
Let \[y=\frac{ax+b}{c}\] i.e., \[y=\frac{a}{c}x+\frac{b}{c}\] i.e., \[y=Ax+B\], where \[A=\frac{a}{c}\],\[B=\frac{b}{c}\] \ \[\bar{y}=A\bar{x}+B\] \ \[y-\bar{y}=A(x-\bar{x})\] Þ \[{{(y-\bar{y})}^{2}}={{A}^{2}}{{(x-\bar{x})}^{2}}\] Þ \[\sum {{(y-\bar{y})}^{2}}={{A}^{2}}\sum {{(x-\bar{x})}^{2}}\] Þ \[n.\sigma _{y}^{2}={{A}^{2}}.n\sigma _{x}^{2}\] Þ \[\sigma _{y}^{2}={{A}^{2}}\sigma _{x}^{2}\] Þ \[{{\sigma }_{y}}=\,|A|{{\sigma }_{x}}\] Þ \[{{\sigma }_{y}}=\,\left| \frac{a}{c} \right|{{\sigma }_{x}}\] Thus, new S.D.\[=\left| \frac{a}{c} \right|\,\sigma \].
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