If for a set of data, \[ \sum_{i=1}^n (x_i - 7) = -8 \quad \text{and} \quad \sum_{i=1}^n (x_i + 3) = 72, \] then find the values of \(\bar{x}\) (the mean) and \(n\) (the number of data points).

Q.If for a set of data, \[ \sum_{i=1}^n (x_i - 7) = -8 \quad \text{and} \quad \sum_{i=1}^n (x_i + 3) = 72, \] then find the values of \(\bar{x}\) (the mean) and \(n\) (the number of data points).

\[ \sum_{i=1}^n (x_i - 7) = -8 \] That is, \[ \sum_{i=1}^n x_i - \sum_{i=1}^n 7 = -8 \] \[ \Rightarrow \sum_{i=1}^n x_i - 7n = -8 \quad \text{—— (i)} \] \[ \sum_{i=1}^n (x_i + 3) = 72 \] That is, \[ \sum_{i=1}^n x_i + \sum_{i=1}^n 3 = 72 \] \[ \Rightarrow \sum_{i=1}^n x_i + 3n = 72 \quad \text{—— (ii)} \] Subtracting equation (i) from equation (ii), we get: \[ 10n = 80 \] \[ \Rightarrow n = 8 \] Substituting \(n = 8\) into equation (i): \[ \sum_{i=1}^n x_i - 56 = -8 \] \[ \Rightarrow \sum_{i=1}^n x_i = 48 \] \[ \therefore \bar{x} = \frac{\sum_{i=1}^n x_i}{n} = \frac{48}{8} = 6 \] Hence, \[ \bar{x} = 6 \quad \text{and} \quad n = 8 \]