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Chapter 18: Problem 727

For observations \(x_{1}, x_{2}, \ldots \ldots . . x_{n}, n_{i=1}(x i+4)=100\) and \(n_{i=1}(x i+6)=140\) then \(n=\) and \(\underline{x}=\) (a) 3,20 (b) 20,3 (c) 1,20 (d) 20,1

Short Answer

Expert verified
The short answer based on the provided step-by-step solution is: (d) 20,1.

Step by step solution

01

Rewrite given observations equations

We have two equations with the summation notation: 1. \(n_{i=1}(x_i+4)=100\) 2. \(n_{i=1}(x_i+6)=140\) Firstly, we rewrite the equations as sums: 1. \(\sum_{i=1}^n (x_i + 4) = 100\) 2. \(\sum_{i=1}^n (x_i + 6) = 140\)
02

Split the sum into simpler terms

Next, we split the sums into sums of xi terms and constants: 1. \(\sum_{i=1}^n x_i + 4n=100\) 2. \(\sum_{i=1}^n x_i + 6n=140\)
03

Solve the system of equations to find n

Now, we can treat these two equations as a system of equations and solve for n. We can subtract equation 1 from equation 2: \((\sum_{i=1}^n x_i + 6n) - (\sum_{i=1}^n x_i + 4n) = 140 - 100\) This simplifies to: \(2n = 40\) Now we solve for n: \(n = \frac{40}{2} = 20\)
04

Calculate the value of \(\sum_{i=1}^n x_i\)

We now substitute the value of n in one of the equations, let's use equation 1: \(\sum_{i=1}^n x_i + 4(20) = 100\) This simplifies to: \(\sum_{i=1}^n x_i + 80 = 100\) Now we solve for the sum of xi values: \(\sum_{i=1}^n x_i = 100 - 80 = 20\)
05

Calculate the mean value \(\underline{x}\)

Finally, we can calculate the mean value \(\underline{x}\) by dividing the sum of xi values by n: \(\underline{x} = \frac{\sum_{i=1}^n x_i}{n} = \frac{20}{20} = 1\) Therefore, the answer is n=20 and \(\underline{x}=1\), which corresponds to option (d).

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