plz help the question from mean and varriance
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Let x and y be the remaining two observations.
Thus,
The observations are 2, 4, 10, 12, 14, x and y.
Given,
Mean = 8
Or (2+4+10+12+14+x+y)/7 = 8
Or x + y + 42 = 56
Or x + y = 14...............(i)
Also,
Variance = 16
Or
∑(xᵢ - Mean) = 16
Or
∑ (xᵢ - 8) = 16
Or ∑ (xᵢ - 8) = 112
Or (2-8)²+(4-8)²+(10-8)²+(12-8)²+(14-8)²+(x-8)²+(y-8)² = 112
Or (-6)²+(-4)²+(2)²+(4)²+(6)²+x² -16x + 64 + y² - 16y + 64 = 112
Or 36 + 16+4+16+36+64+64+ x² + y² -16(x+y) = 112
Or 236 + x² + y² - 16(x+y) = 112
Or x² + y² - 16 × 14 = -124
Or x² + y² - 224 = -124
Or x² + y² = 100..........(ii)
Now,
(i)⇒ x+ y = 14
Or (x+y)² = 14
Or x² + y² + 2xy = 196
Or 100 + 2xy = 196 [From ii]
Or 2xy = 96
Or xy = 48
Or x = 48/y ...............(iii)
Now,
Substituting the value of x in (i), we get
48/y +y = 14
Or 48+y²/y = 14
Or 48 + y² = 14 y
Or y² -14y + 48 = 0
Or y² -(8+6)y + 48 = 0
Or y² -8y -6y + 48 = 0
Or y(y-8) -6(y-8) =0
Or (y-8)(y-6) = 0
Or y = 8 or 6
Now,
(iii)⇒ x = 48/y = 48/8 or 48/6
x = 6 or 8
Thus, 6 and 8 are the remaining observations.
Thus,
The observations are 2, 4, 10, 12, 14, x and y.
Given,
Mean = 8
Or (2+4+10+12+14+x+y)/7 = 8
Or x + y + 42 = 56
Or x + y = 14...............(i)
Also,
Variance = 16
Or
Or
Or ∑ (xᵢ - 8) = 112
Or (2-8)²+(4-8)²+(10-8)²+(12-8)²+(14-8)²+(x-8)²+(y-8)² = 112
Or (-6)²+(-4)²+(2)²+(4)²+(6)²+x² -16x + 64 + y² - 16y + 64 = 112
Or 36 + 16+4+16+36+64+64+ x² + y² -16(x+y) = 112
Or 236 + x² + y² - 16(x+y) = 112
Or x² + y² - 16 × 14 = -124
Or x² + y² - 224 = -124
Or x² + y² = 100..........(ii)
Now,
(i)⇒ x+ y = 14
Or (x+y)² = 14
Or x² + y² + 2xy = 196
Or 100 + 2xy = 196 [From ii]
Or 2xy = 96
Or xy = 48
Or x = 48/y ...............(iii)
Now,
Substituting the value of x in (i), we get
48/y +y = 14
Or 48+y²/y = 14
Or 48 + y² = 14 y
Or y² -14y + 48 = 0
Or y² -(8+6)y + 48 = 0
Or y² -8y -6y + 48 = 0
Or y(y-8) -6(y-8) =0
Or (y-8)(y-6) = 0
Or y = 8 or 6
Now,
(iii)⇒ x = 48/y = 48/8 or 48/6
x = 6 or 8
Thus, 6 and 8 are the remaining observations.
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