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Previous Year Question (PYQs)

Let \(x_i, i = 1,2,\ldots n\) be n observations and \(w_i = p x_i + k\), \(i = 1,2,\ldots n\) where p and k are constants. If the mean of \(x_i\)'s is 48 and standard deviation is 12, whereas the mean of \(w_i\)'s is \(55\) and standard deviation is \(15\), then the values of \(p\) and \(k\) should be:




Solution

For linear transformation \(w = px + k\):

Mean: \(\bar{w} = p\bar{x} + k \Rightarrow 55 = 48p + k\) ... (1)

Standard deviation: \(\sigma_w = |p|\sigma_x \Rightarrow 15 = |p| \times 12\)

\(|p| = 1.25 \Rightarrow p = \pm 1.25\)

If \(p = 1.25\), then \(55 = 48(1.25) + k = 60 + k \Rightarrow k = -5\)

If \(p = -1.25\), then \(55 = 48(-1.25) + k = -60 + k \Rightarrow k = 115\) (not in options)

Therefore \(p = 1.25, k = -5\)


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