The geometric mean of a set of observations is computed as 10. The geometric mean obtained when each observations x iis replaced by \(\rm 3x_i^4\) is
30000
Calculating the Geometric Mean of Transformed Observations
Let's break down this problem involving the geometric mean and a transformation of data points. We are given the geometric mean of an initial set of observations and asked to find the geometric mean of a new set created by applying a specific formula to each original observation.
Understanding the Geometric Mean
The geometric mean (GM) is a type of average that is calculated by multiplying all the numbers in a set and then taking the n-th root of the product, where 'n' is the count of the numbers in the set. It is often used for data that changes multiplicatively, like growth rates.
For a set of \(n\) observations \(x_1, x_2, \dots, x_n\), the geometric mean is given by the formula:
\[ \text{GM}(x) = \sqrt[n]{x_1 \cdot x_2 \cdot \dots \cdot x_n} \]
Initial Geometric Mean
We are given that the geometric mean of the observations \(x_1, x_2, \dots, x_n\) is 10. So, we have:
\[ \sqrt[n]{x_1 \cdot x_2 \cdot \dots \cdot x_n} = 10 \]
Raising both sides of the equation to the power of \(n\), we get the product of the original observations:
\[ x_1 \cdot x_2 \cdot \dots \cdot x_n = 10^n \]
This relationship between the product of the original observations and the given geometric mean will be crucial for solving the problem.
Transformation of Observations
A new set of observations is created by transforming each original observation \(x_i\) into a new observation \(y_i\) using the rule:
\[ y_i = 3x_i^4 \]
So, the new set of observations is \(y_1, y_2, \dots, y_n\), where:
- \(y_1 = 3x_1^4\)
- \(y_2 = 3x_2^4\)
- ...
- \(y_n = 3x_n^4\)
Calculating the Geometric Mean of the New Set
Now, we need to find the geometric mean of the new set of observations \(y_1, y_2, \dots, y_n\). The formula for the geometric mean of the new set is:
\[ \text{GM}(y) = \sqrt[n]{y_1 \cdot y_2 \cdot \dots \cdot y_n} \]
Substitute the expression for \(y_i\) into the formula:
\[ \text{GM}(y) = \sqrt[n]{(3x_1^4) \cdot (3x_2^4) \cdot \dots \cdot (3x_n^4)} \]
Now, let's simplify the expression inside the n-th root:
\[ (3x_1^4) \cdot (3x_2^4) \cdot \dots \cdot (3x_n^4) = (3 \cdot 3 \cdot \dots \cdot 3) \cdot (x_1^4 \cdot x_2^4 \cdot \dots \cdot x_n^4) \]
There are \(n\) terms of '3' being multiplied, and the powers of \(x_i\) can be combined:
\[ (3 \cdot 3 \cdot \dots \cdot 3) \cdot (x_1^4 \cdot x_2^4 \cdot \dots \cdot x_n^4) = 3^n \cdot (x_1 x_2 \dots x_n)^4 \]
So, the geometric mean of the new set becomes:
\[ \text{GM}(y) = \sqrt[n]{3^n \cdot (x_1 x_2 \dots x_n)^4} \]
Using the property of roots \(\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}\) and \(\sqrt[n]{a^k} = a^{k/n}\):
\[ \text{GM}(y) = \sqrt[n]{3^n} \cdot \sqrt[n]{(x_1 x_2 \dots x_n)^4} \]
\[ \text{GM}(y) = 3^{n/n} \cdot (x_1 x_2 \dots x_n)^{4/n} \]
\[ \text{GM}(y) = 3 \cdot (x_1 x_2 \dots x_n)^{4/n} \]
Substituting the Initial Product
We found earlier that \(x_1 \cdot x_2 \cdot \dots \cdot x_n = 10^n\). Substitute this into the expression for \(\text{GM}(y)\):
\[ \text{GM}(y) = 3 \cdot (10^n)^{4/n} \]
Using the power rule for exponents \((a^m)^p = a^{m \cdot p}\):
\[ \text{GM}(y) = 3 \cdot 10^{n \cdot (4/n)} \]
\[ \text{GM}(y) = 3 \cdot 10^4 \]
\[ \text{GM}(y) = 3 \cdot 10000 \]
\[ \text{GM}(y) = 30000 \]
The geometric mean of the new set of observations \(y_i = 3x_i^4\) is 30000.
Summary of Steps
Here is a quick summary of the process:
- Start with the given geometric mean of \(x_i\): \(\text{GM}(x) = 10\).
- Use the GM formula to find the product of \(x_i\): \(x_1 \dots x_n = 10^n\).
- Define the new observations \(y_i = 3x_i^4\).
- Write the GM formula for \(y_i\): \(\text{GM}(y) = \sqrt[n]{y_1 \dots y_n}\).
- Substitute \(y_i = 3x_i^4\) into the formula.
- Simplify the expression using exponent rules.
- Substitute the value of \(x_1 \dots x_n\) found in step 2.
- Calculate the final value of \(\text{GM}(y)\).
| Original Set | Transformation | New Set |
|---|---|---|
| \(x_1, x_2, \dots, x_n\) | \(y_i = 3x_i^4\) | \(y_1, y_2, \dots, y_n\) |
| \(\text{GM}(x) = 10\) | \(\text{GM}(y) = ?\) | |
| \(\sqrt[n]{x_1 \dots x_n} = 10\) | \(\sqrt[n]{y_1 \dots y_n} = ?\) | |
| \(x_1 \dots x_n = 10^n\) | \(y_1 \dots y_n = (3x_1^4) \dots (3x_n^4) = 3^n (x_1 \dots x_n)^4\) | \(\text{GM}(y) = \sqrt[n]{3^n (x_1 \dots x_n)^4} = 3 (x_1 \dots x_n)^{4/n}\) |
| Substitute \(x_1 \dots x_n = 10^n\) | \(\text{GM}(y) = 3 (10^n)^{4/n} = 3 \cdot 10^4 = 30000\) |
Thus, the geometric mean of the new set of observations is 30000.
Revision Table: Geometric Mean Calculation
| Concept | Formula/Rule | Application in Problem |
|---|---|---|
| Geometric Mean (GM) | \(\text{GM}(Z) = \sqrt[n]{z_1 \cdot z_2 \cdot \dots \cdot z_n}\) | Used for both initial \(x_i\) and transformed \(y_i\) sets. |
| Given Information | \(\text{GM}(x) = 10\) | Leads to \(x_1 \dots x_n = 10^n\). |
| Transformation | \(y_i = 3x_i^4\) | Defines the relationship between old and new data. |
| Product of New Observations | \(y_1 \dots y_n = (3x_1^4) \dots (3x_n^4) = 3^n (x_1 \dots x_n)^4\) | Simplification using exponent rules. |
| GM of New Set | \(\text{GM}(y) = \sqrt[n]{y_1 \dots y_n} = \sqrt[n]{3^n (x_1 \dots x_n)^4}\) | Formula for \(\text{GM}(y)\). |
| Simplification of GM(y) | \(\sqrt[n]{a^p b^q} = a^{p/n} b^{q/n}\) | \(\text{GM}(y) = 3 (x_1 \dots x_n)^{4/n}\). |
| Final Calculation | Substitute \(x_1 \dots x_n = 10^n\) | \(\text{GM}(y) = 3 (10^n)^{4/n} = 3 \cdot 10^4 = 30000\). |
Additional Information: Properties of Geometric Mean
The geometric mean has several interesting properties, especially when observations are transformed. Here are a few relevant points:
- Effect of Scaling: If each observation \(x_i\) is multiplied by a constant \(c\), i.e., \(y_i = c x_i\), the new geometric mean is \(c \cdot \text{GM}(x)\).
Proof: \(\text{GM}(y) = \sqrt[n]{cx_1 \cdot cx_2 \dots cx_n} = \sqrt[n]{c^n (x_1 x_2 \dots x_n)} = c \sqrt[n]{x_1 x_2 \dots x_n} = c \cdot \text{GM}(x)\).
- Effect of Power: If each observation \(x_i\) is raised to a power \(k\), i.e., \(y_i = x_i^k\), the new geometric mean is \((\text{GM}(x))^k\).
Proof: \(\text{GM}(y) = \sqrt[n]{x_1^k \cdot x_2^k \dots x_n^k} = \sqrt[n]{(x_1 x_2 \dots x_n)^k} = (x_1 x_2 \dots x_n)^{k/n} = (\sqrt[n]{x_1 x_2 \dots x_n})^k = (\text{GM}(x))^k\).
- Combined Transformation: In our problem, the transformation is \(y_i = 3x_i^4\). This can be seen as a combination of scaling (by 3) and raising to a power (by 4).
Using the properties above: First apply the power \(x_i \to x_i^4\). The GM becomes \((\text{GM}(x))^4\). Then apply the scaling \(x_i^4 \to 3x_i^4\). The GM of \(3x_i^4\) becomes \(3 \cdot (\text{GM}(x))^4\).
Since \(\text{GM}(x) = 10\), the new GM is \(3 \cdot (10)^4 = 3 \cdot 10000 = 30000\). This matches the result obtained through direct calculation, confirming the properties.
These properties can simplify calculations when dealing with transformations involving scaling and powers of observations.
Important Questions from Geometric Progressions
If \(2^{\frac{1}{c}}, 2^{\frac{b}{a c}}, 2^{\frac{1}{a}}\) are in GP, then which one of the following is correct ?
If G is the geometric mean of numbers 1, 2, 22, 23,.....2n-1, then what is the value of 1 + 2log2G ?
If m is the geometric mean of \({\left( {\frac{{\rm{y}}}{{\rm{z}}}} \right)^{\log \left( {{\rm{yz}}} \right)}},{\rm{\;}}{\left( {\frac{{\rm{z}}}{{\rm{x}}}} \right)^{\log \left( {{\rm{zx}}} \right)}}{\rm{\;and\;}}{\left( {\frac{{\rm{x}}}{{\rm{y}}}} \right)^{\log \left( {{\rm{xy}}} \right)}}\) then what is the value of m?
The value of the infinite product \({6^{\frac{1}{2}}} \times {6^{\frac{1}{2}}} \times {6^{\frac{3}{8}}} \times {6^{\frac{1}{4}}} \times \ldots \) is
The geometric mean of the observations x 1, x 2, x 3, … x nis G 1. The geometric mean of the observations y 1, y 2, y 3,… y nis G 2. The geometric mean of observations \(\frac{{{{\rm{x}}_1}}}{{{{\rm{y}}_1}}},\frac{{{{\rm{x}}_2}}}{{{{\rm{y}}_2}}},\frac{{{{\rm{x}}_3}}}{{{{\rm{y}}_3}}}, \ldots \frac{{{{\rm{x}}_{\rm{n}}}}}{{{{\rm{y}}_{\rm{n}}}}}\) is
