13656
Summation Calculation from Mean
This solution details the calculation for a summation expression, using the provided mean of observations.
Step 1: Calculate Sum of Observations
We are given 20 observations (\(x_1, \dots, x_{20}\)) with a mean \(\bar{x}=1.414\). The mean formula is \( \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \).
Using the given values (\(n=20\)):
\( \sum_{i=1}^{20} x_i = n \times \bar{x} = 20 \times 1.414 = 28.28 \)Step 2: Evaluate the Summation Expression
We need to find the value of \(\sum_{i=1}^{20} 100(2x_i + 4)\). We apply summation properties:
1. Factor out the constant 100:
\( 100 \sum_{i=1}^{20} (2x_i + 4) \)2. Split the summation and simplify:
\( 100 \left( \sum_{i=1}^{20} 2x_i + \sum_{i=1}^{20} 4 \right) \)3. Use the sum of \(x_i\) and the sum of a constant (\( \sum_{i=1}^{n} c = n \times c \)):
\( 100 \left( 2 \sum_{i=1}^{20} x_i + (20 \times 4) \right) \)4. Substitute the calculated sum \(\sum_{i=1}^{20} x_i = 28.28\) and calculate:
\( 100 \left( 2 \times 28.28 + 80 \right) \) \( = 100 \left( 56.56 + 80 \right) \) = 100 (136.56) = 13656Final Answer
The calculated value of the summation is 13656.
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