Problem 52 - A First Course in Probability

Question

A population is made up of \(r\) disjoint subgroups. Let \(p_{i}\) denote the proportion of the population that is in subgroup \(i, i=1, \ldots, r .\) If the average weight of the members of subgroup \(i\) is \(w_{i}, i=1, \ldots, r\) what is the average weight of the members of the population?

Step-by-Step Solution

Answer

The average weight of the members of the population is the weighted average given by the formula: \[Average \ Weight \ of \ Population = \sum_{i=1}^r (w_i \times p_i)\]

Step 1: Set up the weighted average formula

The weighted average formula is given by: \[Weighted \ Average = \sum_{i=1}^r (w_i \times p_i) \] where \(r\) is the number of subgroups.

Step 2: Multiply the average weight and proportion for each subgroup

We need to multiply the average weight of each subgroup \(w_i\) with its proportion \(p_i\) for every \(i\) from 1 to \(r\).

Step 3: Compute the sum of the products

Now, find the sum of the products of weights and proportions we calculated in step 2. We will use the summation notation, and sum over all subgroups: \[\sum_{i=1}^r (w_i \times p_i) \]

Step 4: Simplify to find the answer

After finding the sum in step 3, you will get the weighted average which represents the average weight of the entire population. This is our final answer for the problem. So, the average weight of the members of the population is the weighted average given by the formula: \[Average \ Weight \ of \ Population = \sum_{i=1}^r (w_i \times p_i)\]

Key Concepts

Probability Theory: Basics and Its Role in Weighted Averages

Probability theory is a branch of mathematics that handles the analysis of random events. At its core, it provides a way to measure the likelihood of an event occurring. For this reason, probabilities are often expressed as values between 0 and 1, where 0 means the event will not occur and 1 indicates a certainty that it will. In our context of weighted averages, probability theory helps in understanding how likely different outcomes, such as weight ranges within a population, might occur.
A key idea in probability theory is the concept of expected value, which is closely related to weighted averages. This involves summing up all possible outcomes, each multiplied by their probability, to determine the overall average expected outcome. This principle is exactly what we apply while computing the weighted average of population weights. The probability (or proportion) of each subgroup plays a crucial role in determining their contribution to the overall average weight.

Understanding Population Subgroups in Weight Calculations

A population can often be divided into subsets, known as subgroups, that share a common characteristic. For instance, subgroups could be based on factors like age, gender, or health status. These divisions allow us to more accurately study specific segments within a larger group. Understanding these divisions is essential in calculating the average weight using a weighted average approach because each subgroup may have different average weights and proportions.
By identifying these subgroups, we can assign a 'weight' to each, based on their size relative to the total population. This information is signified by their proportion, noted as \(p_i\), within the equation. For example, if one subgroup makes up 30% of the population, this proportion adds to understanding their influence in weighted calculations.

• Subgroups add specificity to calculations.
• They help in effectively applying the weighted average formula.
• Proportions \(p_i\) ensure accurate representation of each subgroup's value.

Comprehending these aspects of population subgroups is crucial, as it ensures the weighted average reflects the true distribution of weights across the entire population.

Calculating Average Weight Using the Weighted Average Formula

The weighted average formula is integral when calculating the average weight, especially in a population of diverse subgroups. It takes into account both the average weight of each subgroup \(w_i\) and their respective size or proportion \(p_i\) within the total population. By doing so, it provides a more representative calculation of the average weight.
To calculate the weighted average weight of a population, follow these simple steps:

• Identify each subgroup's average weight \(w_i\).
• Identify the proportion \(p_i\) of each subgroup within the total population.
• Compute \(w_i \times p_i\) for each subgroup, taking both the average weight and proportion into account.
• Sum all these products for every subgroup: \[\sum_{i=1}^r (w_i \times p_i)\]

This total gives the weighted average, representing the overall average weight of the entire population.
Stick to these steps, and you'll always find the average weight correctly reflecting all subgroups within the population.