STA 506 Homework 2: Sampling Distributions

1 Introduction

A sampling distribution is a probability distribution of a sample statistic calculated from random samples from a population. For this assignment, I will discuss some distributions that are formed from sample statistics and their relationships.

2 Sampling Distributions: A Closer Look

What kinds of statistics can we calculate from a sample? We can calculate means, variances, standard deviations, proportions, and many more. In order to build a sampling distribution, we will need multiple random samples of the same size of n. However, these distributions can looK different depending due to a variety of reasons. For example, what statistic are we interested in? what is the sample size? Do we know what distribution the original population follows? Do we know the population standard deviation? These questions can influence how we go about modeling and using our sampling distributions.

3 Normal Distribution

The normal distribution gets used in specific cases. It typically gets used when we are interested in a sample mean. We typically use a normal distribution when our population of interest is normally distributed, and we know the population variance/standard deviation, and the sample size is large enough \((n \gt 30)\). If we sample from a normally distributed population with a sample size of n, mean \(\mu\) and variance \(\sigma^2\), then our sampling distribution will also follow a normal distribution.

For a sample mean, the distribution will have a mean of \(\mu\), although the variance will now be \(\sigma^2/n\) instead. In other words, if: \[ X_1, X_2, \ldots, X_n \sim N(\mu, \sigma^2)\] then: \[\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)\]

If we standardize our mean, we get: \[ Z = \frac{\bar{X}-\mu}{\sigma/\sqrt{n}} \] As long as the sample size is large enough (or as n \(\to\) \(\infty\)), we can say that \(Z \sim N(0, 1)\)).

4 Student’s T-Distribution

This distribution is very similar to the normal distribution, but it is used when the sample size is small (\(n \leq 30\)) or when the population standard deviation is unknown. When the population standard deviation is unknown, it is estimated with the sample variance, \(S^2\).

We can also standardize the mean with this distribution. If we have a sample: \[ X_1, X_2, \ldots, X_n \sim N(\mu, \sigma^2)\] and the population variance is unknown, then we estimate the variance using the sample variance \[ S^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2. \] And we can obtain: \[ T = \frac{\bar{X}-\mu}{S/\sqrt{n}} \overset{d}{\to} t_{n-1} \] where n-1 is the degrees of freedom for a distribution.

In relation to the normal distribution, as n \(\to \infty\), we can say that \(T \sim N(0, 1)\).

5 Chi-Square Distribution

The Chi-Square (\(\chi^2\)) distribution is yet another distribution that is used for sampling. However, it works differently compared to the normal and t distributions. It is a skewed distribution, and its support is from 0 to \(\infty\). The parameter for this distribution is its degrees of freedom, denoted with either a k or v This distribution is usually used to asymptotically describe sample proportions.

In relation to \(\chi^2\) distributions, when \(Z \sim N(0,1)\), then \(Z^2\) = \(\chi^2(1)\).

6 F Distribution

The F distribution is a distribution used for modeling sample variances. It is usually formed as a ratio of two sample variances, or a ratio of two \(\chi^2\) distributions and their respective degrees of freedom.

To form an F distribution, we can define two \(\chi^2\) random variables, \(U_1\) and \(U_2\), and their respective degress of freedom, \(v_1\) and \(v_2\)