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

We showed that \( n \bar{X}^2 = \frac{1}{n}\left(\sum_{k=1}^{n} X_k\right)^2 \).

Step 1: Express the Sample Mean

First, express the sample mean \( \bar{X} \) of the sample \( \left(X_{1}, X_{2}, \ldots, X_{n} \right) \) using the formula: \[ \bar{X} = \frac{1}{n} \sum_{k=1}^{n} X_{k}. \] This formula gives the average value of the sample.

Step 2: Square the Sample Mean

Next, square the expression for \( \bar{X} \): \[ (\bar{X})^2 = \left( \frac{1}{n} \sum_{k=1}^{n} X_{k} \right)^2. \] This step prepares for comparing it to the other side of the equation.

Step 3: Multiply by the Sample Size

Multiply the squared mean \( (\bar{X})^2 \) by the sample size \( n \):\[ n \bar{X}^2 = n \left( \frac{1}{n} \sum_{k=1}^{n} X_{k} \right)^2. \] Distribute \( n \) across the squared expression.

Step 4: Simplify the Expression

Simplify the expression \[ n \left( \frac{1}{n} \sum_{k=1}^{n} X_{k} \right)^2 = \frac{1}{n} \left( \sum_{k=1}^{n} X_{k} \right)^2. \] This involves canceling the \( n \) in the numerator and denominator, leading to the final expression.

Key Concepts

Sample Size

The "sample size" refers to the number of observations or data points in a given sample. It is denoted by the symbol \( n \). In the context of statistics, the sample size is crucial because it impacts the reliability and accuracy of the statistical results. Generally, a larger sample size leads to more accurate reflections of the population. By increasing the number of data points \( n \), we gain a better approximation of the true population parameters. The formula for the sample mean, \( \bar{X} = \frac{1}{n} \sum_{k=1}^{n} X_{k} \), directly incorporates \( n \) to calculate the average, indicating how fundamental the sample size is in operations like mean calculation.

Squaring a Mean

Squaring a mean involves mathematically raising the mean of the sample, \( \bar{X} \), to the power of two. This action, denoted as \( \bar{X}^2 \), is a crucial step when dealing with sample variance and standard deviation. The process essentially spreads the data over a quadratic form, which is fundamental in variance calculations. Here's how it works:

• Start with the formula for the mean: \( \bar{X} = \frac{1}{n} \sum_{k=1}^{n} X_{k} \).
• Square the entire equation: \( (\bar{X})^2 = \left( \frac{1}{n} \sum_{k=1}^{n} X_{k} \right)^2 \).

Squaring helps to emphasize larger values and reduces the influence of smaller ones. It's a pivotal operation in many statistical computations, such as determining the amount of variability within a sample.

Simplification Steps

Simplification steps are often used to make equations more manageable and to solve them easily. In the context of this exercise, simplification occurs primarily in the third and fourth steps. The goal is to reduce the complexity of the expression involving the sample mean squared, multiplied by the sample size.

• After squaring the mean, multiply by the sample size: \( n \bar{X}^2 = n \left( \frac{1}{n} \sum_{k=1}^{n} X_{k} \right)^2 \).
• Recognize that \( n \times \left( \frac{1}{n} \right)^2 \) can be restructured by canceling \( n \): the result is \( \frac{1}{n} \left( \sum_{k=1}^{n} X_{k} \right)^2 \).

By canceling out the \( n \) values, we simplify the equation to its essence, making the complex fraction cleaner and easier to handle. This prepares the expression for direct application or comparison to other formulas.

Mathematical Proofs

Mathematical proofs are logical arguments that verify the truth of mathematical statements. In this context, the proof aims to demonstrate the validity of the expression \( n \bar{X}^2 = \frac{1}{n} \left( \sum_{k=1}^{n} X_{k} \right)^2 \). This proof involves several key steps:

• Start by defining \( \bar{X} \) as the sample mean.
• Square \( \bar{X} \) to prepare for comparison with the given expression.
• Multiply the squared mean by the sample size \( n \).
• Simplify the expression by canceling terms where possible.

These logical deductions, rooted in algebraic manipulations, uphold the statement's integrity. Proofs such as these are crucial as they lay the groundwork for broader applications and understanding in mathematics, confirming that calculations align with theoretical expectations.