### Unit 3: Sampling Distributions

The concept of sampling distribution lies at the very foundation of statistical inference. It is best to introduce sampling distribution using an example here. Suppose you want to estimate a parameter of a population, say the population mean. There are two natural estimators: 1. sample mean, which is the average value of the data set; and 2. median, which is the middle number when the measurements are arranged in ascending (or descending) order. In particular, for a sample of even size n, the median is the mean of the middle two numbers. But which one is better, and in what sense? This involves repeated sampling, and you want to choose the estimator that would do better on average. It is clear that different samples may give different sample means and medians; some of them may be closer to the truth than the others. Consequently, we cannot compare these two sample statistics or, in general, any two sample statistics on the basis of their performance with a single sample. Instead, you should recognize that sample statistics are themselves random variables; therefore, sample statistics should have frequency distributions by taking into account all possible samples. In this unit, you will study the sampling distribution of several sample statistics. This unit will show you how the central limit theorem can help to approximate sampling distributions in general.

**Completing this unit should take you approximately 5 hours.**

Upon successful completion of this unit, you will be able to:

- apply the central limit theorem to approximate sampling distributions;
- describe the role of sampling distributions in inferential statistics;
- interpret and create graphs of a probability distribution for the mean of a discrete variable;
- describe a sampling distribution in terms of repeated sampling;
- define and compute the mean and standard deviation of the sampling distribution of population proportion p;
- identify or approximate a sampling distribution based on the properties of the population;
- compare and evaluate the sampling distributions of different sample sizes; and
- compare and evaluate the performance of different estimators based on their sampling distributions.

### 3.1: The Concept of Sampling Distributions

### 3.1.1: Continuous Random Variables

First, this section talks about how to describe continuous distributions and compute related probabilities, including some basic facts about the normal distribution. Then, it covers how to compute probabilities related to any normal random variable and gives examples of using -score transformations. Finally, it defines tail probabilities and illustrates how to find them.

### 3.1.2: Definition and Interpretation

This section introduces sampling distribution using a concrete, discrete example, followed by a continuous example. This section also discusses sampling distributions' relationship to inferential statistics.

### 3.1.3: Sampling Distributions Properties

Use the information provided on the demonstration pages and interact with the various simulations.

If you have not done so already, you will need to download install the free Wolfram CDF Player™. If using Chrome as your browser, you will also need to download the CDF files from the pages linked to above, and run them through the CDF Player on your desktop. Other browsers will allow you to interact with the demonstrations directly on the webpage.

### 3.2: Sampling Distributions for Common Statistics

### 3.2.1: The Sampling Distribution of Sample Mean

First, this section discusses the mean and variance of the sampling distribution of the mean. It also shows how central limit theorem can help to approximate the corresponding sampling distributions. Then, it talks about the properties of the sampling distribution for differences between means by giving the formulas of both mean and variance for the sampling distribution. Using the central limit theorem, it also talks about how to compute the probability of a difference between means being beyond a specified value.

This section gives several concrete examples of calculating the exact distributions of the sample mean. The corresponding means and standard deviations are computed for demonstration based on these distributions. Next, it discusses sampling distributions of sample means when the sample size is large. It also considers the case when the population is normal. Finally, it uses the central limit theorem for large sample approximations.

Watch these videos, which discuss sampling distributions.

### 3.2.2: The Sampling Distribution of Pearson's r

### Unit 3 Assessment

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