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  • Let x1, x2, ... xn be n observations. Let wi = lxi + k for i = 1, 2, ...n, where l and k are constants. If the mean of xi’s is 48 and their standard deviation is 12, the mean of wi’s is 55 and standard deviation of wi’s is 15, the values of l and k sh

Let x\textsubscript{1}, x\textsubscript{2}, ... x\textsubscript{n} be n observations. w\textsubscript{i} = lx\textsubscript{i} + k for i = 1, 2, ...n, where l and k are constants. If the mean of x\textsubscript{i}'s is 48 and their standard deviation is 12, the mean of w_i’s is 55 and standard deviation of w_i’s is 15, the values of l and k should be
A. l = 1.25, k = – 5
B. l = – 1.25, k = 5
C. l = 2.5, k = – 5
D. l = 2.5, k = 5

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\\ Given~x_{1},~x_{2} \ldots ..x_{n}\text{ be n observations } \\ \\ \text{And mean of these n observations, }\overline{x}=48~ \\ \\ \text{And their standard deviation S}D_{x}=12~ \\ \\ \text{Another series of n observations is given such that }w_{i}=lx_{i}+k~~for i=1,2 , \ldots .n \text{where L and k are constants} \\ \\ \text{ And mean of these n observations, }\overline{w}=55~ \text{And their standard deviation} SD_{w}=15 \\ \\ \text{ Applying the given condition for mean we get }w_{i}=lx_{i}+k~ \\ \\ \text{Substituting~the corresponding given values of means, we get }55=l \left( 48 \right) +k \ldots \ldots \left( i \right) \\
\\ \text{ We know that if standard deviation of x series is s, then standard deviation of kx series is ks } \\ \\ \text{ So standard deviation of }x_{1},~x_{2}, \ldots .x_{n}\text{is S}D_{x}\text{~ And hence the SD of l}x_{1}\text{, l}x_{2} \ldots ..lx_{n}\text{ is lS}D_{x}~ \\ \\ \text{Similarly, If standard deviation of x series is s, then standard deviation of k+x series is s,} \\ \\ \text{ So S.D of l}x_{1}\text{, l}x_{2} \ldots . lx_{n}\text{is lS}D_{x} \\ \\ \text{And hence the SD of l}x_{1}+k, lx_{2}+k \ldots ..lx_{n}+k is lSD_{x}~ \\ \\ \text{ So applying the given condition for standard deviation we get S}D_{w}=lSD_{x}~ \\ \\ \text{Substituting the given values we get }15=l \left( 12 \right) ~ l=\frac{15}{12}=1.25 \\ \\ \text{ Now substituting the value of l in equation } \left( i \right) ,~\text{we get} 55= \left( 1.25 \right) \left( 48 \right) +k \\ \\ ~ 55=60+k \\ \\ ~ k=-5 \\

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