Multiple choice

A data set of $\displaystyle n$ observations has mean $\displaystyle 2\overset { - }{ x } $ while another data set of $\displaystyle 2n$ observations has mean $\displaystyle \overset { - }{ x } $. The mean of the combined data set of $\displaystyle 3n$ observations will be equal to:

  1. $\displaystyle \overset { - }{ x } $
  2. $\displaystyle \frac { 3 }{ 2 } \overset { - }{ x } $
  3. $\displaystyle \frac { 2 }{ 3 } \overset { - }{ x } $
  4. $\displaystyle \frac { 4 }{ 3 } \overset { - }{ x } $
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D Correct answer
Explanation

Sum of first set = n * 2x_bar. Sum of second set = 2n * x_bar. Total sum = 2n*x_bar + 2n*x_bar = 4n*x_bar. Total observations = 3n. Combined mean = 4n*x_bar / 3n = 4/3 * x_bar.

AI explanation

Using the combined mean formula, we find the total sum of all observations by adding the sums of both groups. The sum of the first data set is n times 2x bar, which is 2nx bar, and the sum of the second is 2n times x bar, which is 2nx bar. The combined sum is 4nx bar, and dividing this by the total of 3n observations gives a new mean of 4/3 times x bar.

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