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Question
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 should be ______.
Options
l = 1.25, k = – 5
l = – 1.25, k = 5
l = 2.5, k = – 5
l = 2.5, k = 5
Solution
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 should be l = 1.25, k = – 5.
Explanation:
Given that `w_i = x_i + k, barx_i = 48`, S.D.`(x_i) = 12`
`w_i` = 55 and S.D.`(w_i)` = 15
Then `barx_i = barxx_i + k` .....(`barw_i` mean of `w_i"'"s` and `barx_i` is the mean of `x_i"'"s`)
⇒ 55 = 48 + k .....(i)
S.D. of wi = S.D. of xi
15 = `l xx 12`
⇒ `l = 15/12` = 1.25 ......(ii)
From eq. (i) and (ii) we have
`k = w_i - barx_i = 55 - 1.25 xx 48`
= 55 – 60
= – 5
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