Correct Answer - Option 1 :
\(\frac{4}{3}\bar x\)
Given
A data set of n observations has mean \(2\bar x\)
Another data set of 2n observations has mean \(\bar x\)
Formula used
Mean = Sum of observations/Number of observations
Calculations
A data set of n observations has mean \(2\bar x\)
\(⇒ 2\bar x = {Sum_1 \over n}\)
⇒ Sum1 = 2̅xn
Another data set of 2n observations has mean \(\bar x\)
\(⇒ \bar x= {Sum_2 \over 2n}\)
⇒ Sum2 = 2̅xn
⇒ Mean of 3n observations = (Sum1 + Sum2)/3n
\(\Rightarrow {2\bar xn + 2\bar xn \over 3n} = {4\bar xn \over 3n}\)
\(\Rightarrow Mean = {4 \over 3}\bar x\)