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.

Asked by Abhisek | 11 months ago |  88

##### Solution :-

Let the remaining two observations be x and y.

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

$$Mean,\overline X$$

$$=\dfrac{6+ 7 +10 +12 +12 +13+x+y}{8}=9$$

60 + x + y= 72

x + y = 12 ...........(1)

Variance = $$\dfrac{1}{n} \displaystyle\sum_{i=1}^{n} f_i (x_i-\overline {x} )^2$$

9.25 = $$\dfrac{1}{8}[(-3)^2 + (-2)^2+1^2+3^2+4^2+x^2+y^2$$

$$-18(x+y)+2\times (9)^2]$$

9.25 =$$\dfrac{1}{8}[9 +4+1+9+9+x^2+y^2$$

$$-18\times 12+162]$$

9.25 = $$\dfrac{1}{8}[48+x^2+y^2-216+162]$$

9.25 = $$\dfrac{1}{8}[x^2+y^2-6]$$

x+ y2=80 ................(2)

From (1), we obtain

x2+y+ 2xy=144 ................(3)

From (2) and (3), we obtain

2xy = 64 ….....................(4)

Subtracting (4) from (2), we obtain

x+ y-2xy = 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.

Answered by Pragya Singh | 11 months ago

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