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Question
The mean of 5 observations is 4.8 and the variance is 6.56. If three of the five observations are 1, 3 and 8, find the other two observations
Solution
Given: n = 5, `bar(x)` = 4.8
Let a, b be the other two observations.
∴ `bar(x) = (sumx_1)/"n" = (1 + 3 + 8 + "a" + "b")/5`
∴ 4.8 = `(12 + "a" + "b")/5`
∴ 24 = 12 + a + b
∴ a + b = 12 ...(1)
Variance = 6.56
∴ 6.56 = `(sumx_1^2)/"n" - (barx)^2`
= `(1 + 9 + 64 + "a"^2 + "b"^2)/5 - (4.8)^2`
= `(74 + "a"^2 + "b"^2)/5 - 23.04`
∴ 29.6 = `(74 + "a"^2 + "b"^2)/5`
∴ 148 = 74 + a2 + b2
∴ a2 + b2 = 74
∴ a2 + (12 – a)2 = 74 ...[By (1)]
∴ a2 + 144 – 24a + a2 = 74
∴ 2a2 – 24a + 70 = 0
∴ a2 – 12a + 35 = 0
∴ (a – 5)(a – 7) = 0
∴ a = 5 or a = 7
∴ b = 7 or b = 5 ...[∵ a + b = 12]
∴ the required observations are 5, 7
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