Chapter 7: Sampling Distribution of the Sample Mean

Welcome to Professor Baker's Math Class! Today, we're tackling Chapter 7, which focuses on understanding the sampling distribution of the sample mean. This is a foundational concept in statistics, allowing us to make inferences about a population based on sample data. Let's break it down!

7.1 Sampling Error: The Need for Sampling Distributions

Sampling error is the error that results from using a sample to estimate a population characteristic. Remember, a sample is just a subset of the entire population, so it's unlikely to perfectly reflect the whole. For instance, consider the IRS reporting the mean tax from a sample of tax returns. Should we expect the mean tax $\bar{x}$ of the sampled returns to be exactly the same as the mean tax $\mu$ of *all* returns? Probably not! This difference is due to sampling error.

7.2 The Mean and Standard Deviation of the Sample Mean

Let's define some terms to make sure we're all on the same page:

  • $\bar{x}$: Represents the sample mean.
  • $\mu$: Represents the population mean.

Key relationships to remember:

  • The mean of the sampling distribution of the sample mean ($\mu_{\bar{x}}$) equals the population mean ($\mu$). This is incredibly important! Even though individual samples will vary, the average of all possible sample means will equal the true population mean.
  • The standard deviation of the sampling distribution of the sample mean ($\sigma_{\bar{x}}$), also known as the standard error, is calculated as:

$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$

Where:

  • $\sigma$ is the population standard deviation
  • $n$ is the sample size

7.3 The Sampling Distribution of the Sample Mean

Definition 7.2: Sampling Distribution of the Sample Mean

For a variable $x$ and a given sample size, the distribution of the variable $\bar{x}$ is called the sampling distribution of the sample mean. This distribution is crucial for understanding how sample means vary and for making probabilistic statements about the population mean.

Key Fact 7.2: Sampling Distribution of the Sample Mean for a Normally Distributed Variable

Suppose that a variable $x$ of a population is normally distributed with mean $\mu$ and standard deviation $\sigma$. Then, for samples of size $n$, the variable $\bar{x}$ is also normally distributed and has mean $\mu$ and standard deviation $\sigma / \sqrt{n}$.

What does it mean? For a normally distributed variable, the possible sample means for samples of a given size are also normally distributed.

Example: Living Space of Homes

Suppose the mean living space for single-family detached homes is 1742 sq. ft. ($\mu = 1742$) with a standard deviation of 568 sq. ft. ($\sigma = 568$).

  • For samples of 25 homes ($n = 25$): $\sigma_{\bar{x}} = \frac{568}{\sqrt{25}} = 113.6$

Important Observations

  • The larger the sample size ($n$), the smaller the standard deviation of $\bar{x}$ ($\sigma_{\bar{x}}$).
  • The smaller the standard deviation of $\bar{x}$, the more closely the possible values of $\bar{x}$ (the possible sample means) cluster around the mean of $\bar{x}$.

Keep practicing, and you'll master these concepts in no time! Good luck, and see you in the next lesson!