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NCERT Class XI Mathematics - Statistics - Solutions

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Question : 28 of 34
Marks: +1, -0
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.
Solution:  
Let the remaining two observations be x and y. Then we are given that 6+7+10+12+12+13+x+y8\frac{6+7+10+12+12+13+x+y}{8} = 9
⇒ 60 + x + y = 72 ⇒ x + y = 12 ... (i)
Also,
18(62+72+102+122+122+132+x2+y2)−(9)2\frac{1}{8}(6^2+7^2+10^2+12^2+12^2+13^2+x^2+y^2)-(9)^2
= 9.25
⇒
18(36+49+100+144+144+169+x2+y2)−81\frac{1}{8}(36+49+100+144+144+169+x^2+y^2)-81
= 9.25
⇒ 642 + x2+y2x^2+y^2 = 722
⇒ x2+y2x^2+y^2 = 80 ... (ii)
Now (x+y)2+(x−y)2(x+y)^2+(x-y)^2 = 2(x2+y2)2(x^2+y^2)
⇒ (12)2+(x−y)2(12)^2+(x-y)^2 = 2 × 80 [Using (i) & (ii)]
⇒ (x−y)2(x-y)^2 = 160 – 144 ⇒ (x−y)2(x-y)^2 = 16 ⇒ x – y = ± 4
When x – y = 4 and x + y = 12, we get x = 8 and y = 4
When x – y = – 4 and x + y = 12, we get x = 4 and y = 8.
So, the remaining two observations are 4 and 8.
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