If \(\overline{X}_1\) and \(\overline{X}_2\) are the means of two distributions with \(\overline{X}_1 < \overline{X}_2\), and \(\overline{X}\) is the mean of the combined distribution, then: The combined mean \(\overline{X}\) is given by: \[ \overline{X} = \frac{n_1\overline{X}_1 + n_2\overline{X}_2}{n_1 + n_2} \] where \(n_1\) and \(n_2\) are the number of observations in the two distributions. \[ Key Points: \] - Since \(\overline{X}_1 < \overline{X}_2\), the combined mean \(\overline{X}\) will always lie between \(\overline{X}_1\) and \(\overline{X}_2\). - The exact position of \(\overline{X}\) depends on the sizes \(n_1\) and \(n_2\): - If \(n_1 = n_2\), then \(\overline{X}\) is exactly the average of \(\overline{X}_1\) and \(\overline{X}_2\). - If \(n_1 > n_2\), \(\overline{X}\) will be closer to \(\overline{X}_1\). - If \(n_2 > n_1\), \(\overline{X}\) will be closer to \(\overline{X}_2\). \[ Final Result: \] \[ \overline{X}_1 < \overline{X} < \overline{X}_2 \]