The mean and variance of eight observations are 9 and 9.25, respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.

Let the
remaining two observations be *x* and* y*.

Therefore,
the observations are 6, 7, 10, 12, 12, 13, *x*, *y*.

From (1), we obtain

*x*^{2}
+ *y*^{2} + 2*xy* = 144
…(3)

From (2) and (3), we obtain

2*xy *=
64 … (4)

Subtracting (4) from (2), we obtain

*x*^{2}
+ *y*^{2 }– 2*xy* = 80 – 64 = 16

⇒ *x*
– *y* = ±
4 … (5)

Therefore, from (1) and (5), we obtain

*x* =
8 and *y* = 4, when *x* – *y* = 4

*x* =
4 and *y* = 8, when *x* – *y* = –4

Thus, the remaining observations are 4 and 8.

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