Problem 33 - Basic Statistics for Business and Economics with Student

Question

In 2001 there were 42.0 million pager subscribers. By 2006 the number of subscribers increased to 70.0 million. What is the geometric mean annual increase for the period?

Step-by-Step Solution

Answer

Geometric mean annual increase is 10.7%.

Step 1: Understand the Problem

First, identify that we are asked to find the geometric mean annual increase between 2001 and 2006. This means we need to calculate the constant percentage increase that resulted in the growth from 42.0 million to 70.0 million over 5 years.

Step 2: Geometric Mean Formula

The geometric mean annual increase can be found by using the formula: \((\text{Final Value} / \text{Initial Value})^{1/n} - 1\), where \(n\) is the number of periods (years).

Step 3: Apply Values to Formula

Substitute the values into the formula: \((70/42)^{1/5} - 1\). The initial value is 42.0 million, the final value is 70.0 million, and the number of years \(n\) is 5.

Step 4: Compute the Ratio

Calculate the ratio \(70/42\), which simplifies to approximately 1.6667.

Step 5: Calculate the 5th Root

Calculate \(1.6667^{1/5}\) using a calculator to find the fifth root, which is approximately 1.107.

Step 6: Subtract 1 to Find the Increase Rate

Now subtract 1 from 1.107 to find the annual increase rate: \(1.107 - 1 = 0.107\).

Step 7: Convert to Percentage

Convert the decimal to a percentage by multiplying by 100. Thus, the geometric mean annual increase is approximately \(10.7\%\).

Key Concepts

Growth Rate Calculation

Understanding the growth rate calculation is essential when analyzing changes over a specific period. It helps us determine the steady rate at which something, such as a company's sales or subscription numbers, increases or decreases annually over time.
In our example, we considered the growth of pager subscribers from 2001 to 2006, growing from 42 million to 70 million. The main goal is to find the constant annual percentage increase that represents this total growth over the given timeframe.
The growth rate calculation requires using the geometric mean formula, which stabilizes the variations into a constant growth rate for each year. This approach is crucial for assessing long-term investments and trends accurately.

Percentage Increase

Percentage increase is a way to express the amount of growth as a percentage of the starting value. It provides a clear idea about the rate of change over time.
In this problem, we calculated the percentage increase each year over five years. It shows us by what percentage the number of subscribers has grown annually, providing a simple way to communicate the rate of growth.
By subtracting 1 from the fifth root of the ratio of final to initial subscribers, and then converting this into a percentage, we've found that there was a 10.7% average annual increase. This percentage increase is a helpful metric for comparing different growth rates across various sectors or time periods, allowing for easier evaluation and analysis.

Geometric Mean Formula

The geometric mean formula is used to calculate an average growth rate over a series of periods. Unlike an arithmetic average, which adds values together and divides by the number of values, the geometric mean considers compounding, which makes it more suitable for growth rates.
In our situation, the formula used is \[ \left( \frac{\text{Final Value}}{\text{Initial Value}} \right)^{1/n} - 1 \]where the final value is 70 million subscribers, the initial value is 42 million, and \( n \) represents the 5 year period.
Applying this formula gives a more accurate representation of annual growth as it accounts for the compounding effect over time. This effect is vital in understanding the real financial implications of growth, whether we are talking about investments, populations, or market shares.
The geometric mean illuminates the constant rate of growth in a more realistic manner compared to linear growth representations, allowing for a precise interpretation of how the value is expanding each period.