Chapter 7 – Sampling Distributions (STAT 201: Elementary Statistics)

Sampling Distributions

Introduction to Sampling Distributions

Sampling distributions are a fundamental concept in statistics, describing the probability distribution of a given statistic based on a random sample. Understanding sampling distributions allows statisticians to make inferences about population parameters from sample statistics.

• Population: The entire group of individuals or items of interest.

• Sample: A subset of the population selected for analysis.

• Statistic: A numerical value calculated from sample data (e.g., sample mean 𝑥, sample proportion 𝕣).

• Parameter: A numerical value that describes a characteristic of the population (e.g., population mean μ, population proportion p).

Purpose of Statistics

The main purpose of statistics is to use sample data to make predictions or inferences about population parameters. Since it is often impractical to study every member of a population, samples are used to estimate population characteristics.

• Sample Mean (𝑥): Used to predict the population mean (μ) for quantitative variables.

• Sample Proportion (𝕣): Used to predict the population proportion (p) for categorical variables.

• Sampling Variability: Sample statistics can vary from sample to sample due to random selection.

Examples of Sampling

Consider a population of size 6. If we randomly select samples of size 2, there are 15 possible samples. The sample means from these samples may or may not match the population mean.

• Number of possible samples: $\frac{6!}{2!(6-2)!} = 15$

• Sample Means: Only some sample means will exactly equal the population mean; others will differ.

Types of Distributions

There are three main types of distributions in statistics:

• Population Distribution: Distribution of all data values in the population.

• Sample Distribution: Distribution of data values within a single sample.

• Sampling Distribution: Distribution of a statistic (e.g., mean, proportion) calculated from all possible samples of a given size from the population.

Properties of Sampling Distributions

The mean of the sampling distribution of a statistic is equal to the corresponding population parameter.

• Mean of Sample Means: $\mu_{\bar{x}} = \mu$

• Mean of Sample Proportions: $\mu_{\hat{p}} = p$

• Key Point: The mean of the sampling distribution does not change with sample size.

Standard Error of Sampling Distributions

The standard error measures the typical distance that a sample statistic falls from the population parameter. It quantifies the variability of the statistic across different samples.

• Standard Error of the Mean: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ Represents the standard deviation of sample means.

• Standard Error of the Proportion: $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$ Represents the standard deviation of sample proportions.

• Effect of Sample Size: As sample size increases, standard error decreases; as sample size decreases, standard error increases.

Normality of Sampling Distributions

Under certain conditions, sampling distributions are approximately normal, which allows for the use of normal probability calculations.

• Central Limit Theorem (CLT): For sample means, if the sample size is large enough (typically $n \geq 30$), or if the population is normally distributed, the sampling distribution of the sample mean is approximately normal.

• Normal Approximation for Proportions: The sampling distribution of the sample proportion is approximately normal if $np \geq 15$ and $n(1-p) \geq 15$.

Applications and Examples

Sampling distributions are used to:

• Determine if a sample statistic is unusual.

• Calculate probabilities for a range of sample values.

• Make inferences about population parameters based on sample data.

Example Table: Conditions for Normality

Statistic	Condition for Normality
Sample Mean	$n \geq 30$ or population is normal
Sample Proportion	$np \geq 15$ and $n(1-p) \geq 15$

Calculating Probabilities Using Sampling Distributions

Probabilities can be calculated for sample means or proportions using the normal distribution, provided the conditions for normality are met.

• For Means: Use $z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}}$

• For Proportions: Use $z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}$

• Find the area under the normal curve using z-tables to determine probabilities.

Example Table: Formulas for Sampling Distributions

Statistic	Mean	Standard Error
Sample Mean	$\mu_{\bar{x}} = \mu$	$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$
Sample Proportion	$\mu_{\hat{p}} = p$	$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

Summary

• Sampling distributions describe the variability of sample statistics.

• The mean of the sampling distribution equals the population parameter.

• Standard error quantifies the spread of the sampling distribution and decreases with increasing sample size.

• Normality conditions allow for probability calculations using the normal distribution.

Additional info: These notes expand on the brief points and diagrams in the original slides, providing full definitions, formulas, and context for exam preparation.