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A data set of n observations has mean \(2\bar x\) while another data set of 2n observations has mean \(\bar x\). The mean of the combined data set of 3n observations will be equal to
1. \(\frac{4}{3}\bar x\)
2. \(\frac{3}{2}\bar x\)
3. \(\frac{2}{3}\bar x\)
4. None of the above

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Best answer
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

\(\Rightarrow {2\bar xn + 2\bar xn \over 3n} = {4\bar xn \over 3n}\)

\(\Rightarrow Mean = {4 \over 3}\bar x\)

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