Correct Answer - Option 1 : 30
Given
n1 = 60
n2 = 40
x̅1 = 45
x̅2 = 55
σ21 = 4
σ22 = 9
Formula used
Combined mean = X̅ = (n1 × x̅1 + n2 × x̅ 2)/(n1 + n2)
Variance of combined mean = n1(σ21 + d21) + n2(σ22 + d22)/(n1 + n2)
n1 and n2 are group 1 and group 2 observation
x̅1 and x̅2 are mean of group 1 and group 2
σ21 and σ22 are variance of group 1 and group 2
d1 = x̅1 – X̅
d2 = x̅2 – X̅
Calculation
The combined mean is X̅
⇒ (60 × 45 + 40 × 55)/(60 + 40)
⇒ (2700 + 2200)/100
∴ Combined mean is 49
⇒ d1 = 45 – 49
⇒ d1 = -4
⇒ d21 = 16
⇒ d2 = 55 – 49 = 6
⇒ d22 = 36
The combined variance of both group
⇒ 60(4 + 16) + 40(9 + 36)/(60 + 40)
⇒ (60 × 20 + 40 × 45)/(100)
⇒ (1200 + 1800)/100
⇒ 3000/100
∴ The combined variance is 30