(a) If z1=x1+y1, z2=x2+y2, …zn=xn+yn, then prove that z̅=x̅+y̅, where the symbols have their usu

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(a) If z1=x1+y1, z2=x2+y2, …zn=xn+yn, then prove that...

(a) If $z_{1}=x_{1}+y_{1}, z_{2}=x_{2}+y_{2}, \ldots z_{n}=x_{n}+y_{n}$, then prove that $\bar{z}=\bar{x}+\bar{y}$,
where the symbols have their usual meaning.
(b) Prove that the logarithm of geometric mean of observations is the arithmetic mean of logarithms of the observations.

Key Concepts

Properties of Logarithms

Logarithms have several important properties, including the facts that the logarithm of a product is equal to the sum of the logarithms and that the logarithm of a number raised to a power is equal to that power times the logarithm of the number. These properties are essential in transforming multiplicative relationships into additive ones, as seen in the second part of the question where the logarithm of the geometric mean is expressed as the arithmetic mean of logarithms.

Linearity of Summation

Linearity of summation refers to the property that the sum of the sums is equal to the sum of the individual components. This property allows one to break up sums and distribute operations over them. It is employed in the first part of the problem where the sum over the combined observations is split into separate sums for x and y, which leads directly to the conclusion that the mean of the combined observations equals the sum of the individual means.

Arithmetic Mean

The arithmetic mean is the sum of a set of numbers divided by the count of numbers in the set. It represents a central value of a data set and is often used to summarize the overall behavior of the data. In the context of the problem, the arithmetic mean's linearity property—that the mean of sums equals the sum of the means—is crucial for proving the identity involving z?, x?, and y?.

Geometric Mean

The geometric mean is defined as the nth root of the product of n positive numbers and is used to describe multiplicative or exponential relationships. It is particularly useful when comparing different items that have different properties. In the problem, establishing the logarithmic equivalence, where the log of the geometric mean equals the arithmetic mean of the logarithms, highlights a key relationship between these two means.

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