Problem 12 - Calculus for Biology and Medicine

Question

Let \(\left(X_{1}, X_{2}, \ldots, X_{n}\right)\) denote a sample of size \(n\). Show that $$ n \bar{X}^{2}=\frac{1}{n}\left(\sum_{k=1}^{n} X_{k}\right)^{2} $$ where \(\bar{X}\) is the sample mean.

Step-by-Step Solution

Answer

Multiply the sample mean by \(n\), square it, and divide by \(n\) to verify the equality.

Step 1: Definition of Sample Mean

The sample mean \(\bar{X}\) is defined as the average of all sample observations. Mathematically, it is represented as \(\bar{X} = \frac{1}{n} \sum_{k=1}^{n} X_k\).

Step 2: Expression for \(n \bar{X}\)

Multiply both sides of the sample mean formula by \(n\) to get: \(n \bar{X} = \sum_{k=1}^{n} X_k\).

Step 3: Square Both Sides

Square the equation obtained in Step 2: \((n \bar{X})^2 = \left(\sum_{k=1}^{n} X_k\right)^2\).

Step 4: Expression for \(n \bar{X}^2\)

Divide both sides of the equation from Step 3 by \(n\): \(n \bar{X}^2 = \frac{1}{n} \left(\sum_{k=1}^{n} X_k\right)^2\). This is the required result.

Key Concepts

Understanding Sample Size in Statistics

In statistics, the concept of sample size refers to the number of observations in a sample, which is denoted by the symbol "n".

When conducting statistical analyses, each observation in the sample contributes to the calculations and findings. A larger sample size generally leads to more reliable and accurate results, because it provides a more comprehensive picture of the population. On the other hand, a smaller sample size may lead to less reliable outcomes, as it might not capture all the variability present in the population.

It's crucial to choose an appropriate sample size for your study. This involves considering the objectives of the analysis, the statistical methods being employed, and the nature of the data. By understanding sample size, researchers and analysts can design studies that yield meaningful insights.

Creating Mathematical Expressions

Mathematical expressions are essential in representing complex ideas and calculations in a concise way.

In the exercise we are discussing, one important expression involves the sample mean, which is noted as \(\bar{X}\). The sample mean is calculated by taking the sum of all observed values \(X_k\) and dividing them by the number of observations, \(n\). Thus, it is mathematically expressed as: \[\bar{X} = \frac{1}{n} \sum_{k=1}^{n} X_k\]

This allows researchers to use mathematical language to convey statistical concepts, making the data and results more universally understandable.

Mathematical expressions are not only useful for defining parameters but also for deriving new insights and results from known data, as demonstrated in this exercise.

Following a Step-by-Step Solution Approach

A step-by-step solution approach breaks down complex problems into manageable parts, making it easier to understand and solve them effectively.

Let's go through the solution as given in the exercise:

• **Step 1:** Understand the component you are dealing with, in this case, the sample mean \(\bar{X}\). Define it mathematically: \(\bar{X} = \frac{1}{n}\sum_{k=1}^{n} X_k\).


• **Step 2:** Manipulate the expression by multiplying both sides by \(n\), resulting in: \(n \bar{X} = \sum_{k=1}^{n} X_k\).


• **Step 3:** To reveal the next piece of the solution, square both sides of the previous equation: \((n \bar{X})^2 = (\sum_{k=1}^{n} X_k)^2\).


• **Step 4:** Finally, divide by \(n\) to achieve the target expression: \(n \bar{X}^2 = \frac{1}{n}(\sum_{k=1}^{n} X_k)^2\).

Each step logically follows from the previous one, illustrating how to transform expressions while preserving equality. A step-by-step approach not only helps solve current problems but also equips students with a strategy to tackle future assignments.