Let the Mean and Variance of five observations $ x_i $, $ i = 1, 2, 3, 4, 5 $ be 5 and 10 respectively. If three observations are $ x_1 = 1, x_2 = 3, x_3 = a $ and $ x_4 = 7, x_5 = b $ with $ a>b $, then the Variance of the observations $ n + x_n $ for $ n = 1, 2, 3, 4, 5 $ is
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Use the formulas for mean and variance to set up equations based on the given information. Solve these equations to find the values of the unknown observations. Then, apply the transformation to the original observations to get the new set of observations and calculate their variance.
The mean is given by \( \bar{x} = \frac{\sum x_i}{n} = \frac{1 + 3 + a + 7 + b}{5} = 5 \). This simplifies to \( 11 + a + b = 25 \), and subsequently \( a + b = 14 \). The variance is given by \( \sigma^2 = \frac{\sum x_i^2}{n} - \left( \bar{x} \right)^2 = 10 \). Substituting values, we get \( \frac{1^2 + 3^2 + a^2 + 7^2 + b^2}{5} - (5)^2 = 10 \), which further simplifies to \( \frac{1 + 9 + a^2 + 49 + b^2}{5} - 25 = 10 \). This leads to \( 59 + a^2 + b^2 = 175 \), and \( a^2 + b^2 = 116 \). We have the system of equations: \( a + b = 14 \) and \( a^2 + b^2 = 116 \). Using the identity \( (a + b)^2 = a^2 + b^2 + 2ab \), we substitute the known values: \( 14^2 = 116 + 2ab \). This gives \( 196 = 116 + 2ab \), so \( 2ab = 80 \), and \( ab = 40 \). Solving the system \( a + b = 14 \) and \( ab = 40 \) by forming the quadratic \( t^2 - (a + b)t + ab = 0 \), we get \( t^2 - 14t + 40 = 0 \). Factoring yields \( (t - 10)(t - 4) = 0 \). Thus, \( t = 10 \) or \( t = 4 \). Given \( a>b \), we determine that \( a = 10 \) and \( b = 4 \). The original observations \( x_i \) are \( 1, 3, 10, 7, 4 \). The new observations \( n + x_n \) for \( n = 1 \) to \( 5 \) are: \( 1 + x_1 = 1 + 1 = 2 \), \( 2 + x_2 = 2 + 3 = 5 \), \( 3 + x_3 = 3 + 10 = 13 \), \( 4 + x_4 = 4 + 7 = 11 \), \( 5 + x_5 = 5 + 4 = 9 \). The new set of observations is \( 2, 5, 13, 11, 9 \). The mean of this new set is \( \frac{2 + 5 + 13 + 11 + 9}{5} = \frac{40}{5} = 8 \). The variance of the new set is \( \frac{2^2 + 5^2 + 13^2 + 11^2 + 9^2}{5} - (8)^2 \). This calculates to \( \frac{4 + 25 + 169 + 121 + 81}{5} - 64 = \frac{400}{5} - 64 = 80 - 64 = 16 \).
