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Chapter 18: Problem 701

The average of \(n\) numbers \(y_{1}, y_{2} \ldots \ldots . . y_{n}\) is M. If \(y_{n}\) is replaced by \(\mathrm{y}^{\prime}\) then the new average is (a) \(\left[\left(M+y_{n}-y^{\prime}\right) / n\right]\) (b) \(\left[\left\\{(n-1) M+y^{\prime}\right\\} / n\right]\) (c) \(\left[\left(n M-y_{n}+y^{\prime}\right) / n\right]\) (d) \(M-y_{n}-y^{\prime}\)

Short Answer

Expert verified
The new average when \(y_n\) is replaced by \(y'\) will be (c) \(\left[\left(n M-y_{n}+y^{\prime}\right) / n\right]\).

Step by step solution

01

Find the sum of n numbers

To find the sum of the n numbers, we can multiply the average (M) by n: Sum = M * n
02

Replace yn by y'

Now, we are replacing the number yn with y'. To find the new sum, we can first subtract yn from the sum we calculated in step 1, and then add y': New Sum = Sum - yn + y'
03

Substitute the expression for Sum

From step 1, we can substitute Sum = M * n in step 2: New Sum = (M * n) - yn + y'
04

Calculate the new average

Now, we can find the new average by dividing the new sum by n: New Average = New Sum / n
05

Simplify and find the correct option

Simplify the expression derived in step 4 and compare it with the given options: New Average = [(M * n) - yn + y'] / n = [(n * M - yn + y')] / n Comparing this simplified expression with the given options, we find that the correct answer is (c). So, the new average when yn is replaced by y' will be: (c) \(\left[\left(n M-y_{n}+y^{\prime}\right) / n\right]\)

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Most popular questions from this chapter

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

The mean of the series \(a, a+d, a+2 d \ldots \ldots . a+(2 n+1) d\) is (a) \(a+[(2 n+1) / 2] d\) (b) \(a+(n+1) d\) (c) \(a+(2 n+1) d\) (d) \(a+[(2 n-1) / 2] d\)

For 100 observations \(\sum(x i-30)=0\) and \(\sum(x i-30)^{2}=10000\) then C.V. (coefficient of variance) is \(\%\) (a) 10 (b) 100 (c) \(33.33\) (d) 30

The standard deviation and coefficient of variation of \(7,7,7\), 7,7 is (a) 0,7 (b) 7,0 (c) 7,7 (d) 0,0

Three identical dice are rolled. The probability that the same number will appear on each of them is (a) \((1 / 6)\) (b) \((1 / 36)\) (c) \((1 / 216)\) (d) \((1 / 18)\)
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