Welcome back to class! In this session, we started Chapter 7 by tackling a fundamental concept in statistics: the Sampling Distribution of the Sample Mean. If you have ever wondered how statisticians can make accurate predictions about a massive population using only a relatively small sample, this lesson holds the answer.
Defining the Concept
First, let's look at Definition 7.2. When we take a sample from a population and calculate the mean of that sample, we get a value called $\bar{x}$. If we were to take every possible sample of a specific size ($n$) from that population and calculate the mean for each one, the distribution of those means is what we call the sampling distribution of the sample mean.
The Basketball Team Example
To visualize this, we looked at a small population: the 5 starting players on a men's basketball team. Their heights (in inches) were 76, 78, 79, 81, and 86. The population mean height was calculated as:
$$ \mu = 80 $$We then looked at what happens when we take samples of size $n=2$. By listing every possible pair of players (AB, AC, AD, etc.) and calculating their average heights, we noticed a pattern. While individual sample means varied (ranging from 77 to 83.5), the mean of these sample means ($\%mu_{\bar{x}}$) was exactly equal to the population mean.
As we increased the sample size ($n=1, 2, 3, 4, 5$), the dot plots in our notes revealed a crucial insight: as the sample size increases, the spread of the data decreases. By the time we reached a sample size of 5 (the whole population), the standard deviation dropped to 0.
Key Formulas
This relationship is governed by two very important formulas derived in this chapter:
- Formula 7.1: Mean of the Sample Mean
The mean of the variable $\bar{x}$ equals the mean of the population.
$$ \mu_{\bar{x}} = \mu $$ - Formula 7.2: Standard Deviation of the Sample Mean
The standard deviation of the variable $\bar{x}$ (often called the Standard Error) equals the population standard deviation divided by the square root of the sample size.
$$ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} $$
Putting it into Practice: Living Space
We applied these formulas to a real-world scenario involving the living space of single-family homes. Given a population mean $\mu = 1742$ sq. ft. and a standard deviation $\sigma = 568$ sq. ft., we compared two different sample sizes.
Case A: Sample size of 25
Using our formula, we calculated the standard deviation for the sample mean:
Case B: Sample size of 500
When we increased the sample size significantly, look what happened to the deviation:
This demonstrates the power of large samples! The standard deviation dropped from 113.6 to 25.40. This means that with a larger sample size, our sample mean is likely to be much closer to the true population mean.
Summary
Whether we are looking at basketball players, house sizes, or birth weights (as seen in the final example of the notes), the math remains consistent. Remember that $\mu_{\bar{x}}$ will always center on the population mean, but your precision—measured by $\sigma_{\bar{x}}$—improves drastically as your sample size ($n$) grows. Keep practicing those calculations, and I'll see you in the next class!