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 .

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

Mean,

x = (6 + 7 + 10 + 12 + 12 + 13 + x + y)

9 = (60 + x + y)/8

60 + x + y = 72

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

Variance,

9.25 = 1/n ∑⁸_{i = 1}(x_i - x)²

9.25 = 1/8 [(- 3)² + (-2)² + (1)² + (3)² + (3)² + (4)² + x² + y² - 2 x 9(x + y) + 2 x (9)²]

9.25 = 1/8 [9 + 4 + 1+ 9 + 9 + 16 + x² + y² - 18(12) + 162]

9.25 = 1/8 [48 + x² + y² - 216 + 162]

9.25 = 1/8 [x² + y² - 6]

x² + y² = 80 ....(2)

From (1) , we obtain

x² + y² + 2xy = 144 ....(3)

From (2) and (3) , we obtain

2xy = 64 ....(4)

Subtracting (4) from (2) , we obtain

x² + y² - 2xy = 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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NCERT Solutions Class 11 Maths Chapter 15 Exercise ME Question 1

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.

Summary:

Given that 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. Thus, the remaining two observations are 4 and 8