Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from normal distributed population with mean \(\mu\) and variance \(\sigma^{2}\). Let \(s^{2}\) be the sample variance. Prove that \(\frac{(n-1) s^{2}}{\sigma^{2}}\) has \(\chi^{2}\) distribution with \(n-1\) degrees of freedom.

Short answer

\(\frac{(n-1) s^{2}}{\sigma^{2}}\) has a \(\chi^{2}\) distribution with \(n-1\) degrees of freedom.

Step-by-step solution

Identify Random Sample

Given \(X_{1}, X_{2}, \ldots, X_{n}\) as a random sample from a normally distributed population with mean \(\mu\) and variance \(\sigma^{2}\)

Understand Sample Variance

\(s^{2}\) is given as the sample variance, which is the variance calculated from the given sample. The formula for sample variance is \(s^{2} = \frac{1}{n-1} \sum_{i=1}^{n}(X_{i}-\overline{X})^{2}\)

Apply Given Statistic

We apply the statistic \(\frac{(n-1) s^{2}}{\sigma^{2}}\) and plug in the formula for \(s^{2}\). This gives us \(\frac{(n-1) \sum_{i=1}^{n}(X_{i}-\overline{X})^{2}}{n \sigma^{2}}\)

Simplify the Expression

After simplifying, the expression becomes \(\frac{ \sum_{i=1}^{n}(X_{i}-\overline{X})^2}{\sigma^{2}}\). It can be shown that this is equal to \(\sum_{i=1}^{n} Z_i^2\) where \(Z_i\) are independent standard normal random variables.

Identify Chi-Square Distribution

The sum of squares of independent standard normal random variables have chi-square distribution. Hence the given statistic has a chi-square distribution.

Identify Degrees of Freedom

The degrees of freedom is \(n-1\) as given by the adjustment factor in the statistic. The degrees of freedom for a sample is typically defined as the number of values in the calculation that are free to vary.

Key concepts

Sample Variance

Sample variance is a measure used to quantify the variation or dispersion within a set of sample values. Think of it as an average of how much each data point in your sample differs from the mean (average) of the sample itself. This concept is pivotal in statistics to understand the spread of data points around the mean.

The formula to calculate sample variance, denoted as \(s^2\), is:

• \(s^2 = \frac{1}{n-1} \sum_{i=1}^{n}(X_{i}-\overline{X})^{2}\)

Here, \(X_i\) represents each data point, \(\overline{X}\) is the sample mean, and \(n\) is the number of observations in the sample. Notice the division by \(n-1\) instead of \(n\). This is called Bessel's correction, making \(s^2\) an unbiased estimator of the population variance \(\sigma^2\).

Sample variance is particularly useful in inferential statistics where a finite sample is used to make accurate predictions about a larger population.

Degrees of Freedom

Degrees of freedom might sound complex, but it’s quite an accessible concept. Essentially, it refers to the number of independent values or quantities which can vary in the analysis without breaking any constraints.

In the context of calculating sample variance, degrees of freedom comes into play because when computing the sample mean, one degree of freedom is lost. Therefore, instead of using \(n\) as the denominator in the variance formula, we use \(n-1\). This adjustment provides a more accurate estimation of the population variance by accounting for the freedom lost when estimating the mean.

Understanding degrees of freedom is crucial, especially when dealing with distributions like the chi-square distribution, where it defines the shape of the distribution. In the provided exercise, the degrees of freedom are \(n-1\), which essentially means that \(n-1\) independent pieces of information from the sample contribute to the computation of the variance.

Normal Distribution

The normal distribution, often referred to as the 'bell curve,' is a fundamental concept in statistics. It describes how values of a variable are distributed across a dataset in a way that is symmetrical, with most of the observations clustering around the mean while tapering off equally towards both ends.

Normal distributions are characterized by two key parameters: the mean (µ), which determines the center of the distribution, and the variance (\(\sigma^2\)), which dictates the spread or dispersion of the data. Many variables in the real world follow this distribution, making it a critical foundation for statistical theories and methods, especially in inferential statistics where assumptions about the population distribution are made.

In the exercise, the assumption that the sample comes from a normal distribution with a given mean and variance leads to the application of statistical techniques like the chi-square distribution, ensuring the analysis respects the innate characteristics of the data.