Chapter 10: Estimation - Single Samples

Welcome to Chapter 10! In this chapter, we'll explore the fundamental concepts of estimation using single samples. Estimation allows us to make informed guesses about population parameters based on sample data. We'll cover point estimation, interval estimation for the population mean, and estimating population proportions. Let's get started!

10.1 Point Estimation of the Population Mean

A point estimator is a single value that serves as our "best guess" for a population parameter. For example, the sample mean $\bar{x}$ is a point estimator for the population mean $\mu$.

Here's a quick recap of point estimators:

  • Sample Mean ($\bar{x}$): Estimates the population mean ($\mu$)
  • Sample Proportion ($\hat{p}$): Estimates the population proportion ($p$)
  • Sample Standard Deviation ($s$): Estimates the population standard deviation ($\sigma$)

So, if you calculate the sample mean to be 12.7, then you estimate the population mean to be 12.7 as well. Easy, right?

10.2 Interval Estimation of the Population Mean

While point estimates are useful, they don't give us a sense of the uncertainty associated with our estimate. That's where interval estimation comes in. An interval estimate provides a range of values within which we believe the true population parameter lies.

A confidence interval is a type of interval estimate that is associated with a confidence level (e.g., 95%). A 95% confidence interval means that if we were to repeat our sampling process many times, 95% of the resulting intervals would contain the true population parameter.

When the population standard deviation ($\sigma$) is known, we can calculate a confidence interval for the population mean ($\mu$) using the following formula:

$$ \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}} $$

Where:

  • $\bar{x}$ is the sample mean
  • $z_{\alpha/2}$ is the critical value from the standard normal distribution corresponding to the desired confidence level
  • $\sigma$ is the population standard deviation
  • $n$ is the sample size

For example, to construct a 95% confidence interval, $z_{\alpha/2}$ would be 1.96. This value represents the number of standard deviations away from the mean that captures 95% of the area under the standard normal curve.

Example: A random sample of 100 car engines has a mean weight of 425 pounds. Assume the population standard deviation is 900 pounds. Construct a 95% confidence interval for the population mean.

$$425 \pm 1.96 * \frac{900}{\sqrt{100}}$$

$$425 \pm 176.4$$

Therefore, the 95% confidence interval is (248.6, 601.4)

10.3 Estimating the Population Proportion

Sometimes, we're interested in estimating the proportion of a population that possesses a certain characteristic. For example, we might want to estimate the proportion of voters who support a particular candidate.

We can estimate the population proportion (p) using the sample proportion ($\hat{p}$). To construct a confidence interval for $p$, we can use the following formula:

$$ \hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$

Where:

  • $\hat{p}$ is the sample proportion
  • $z_{\alpha/2}$ is the critical value from the standard normal distribution
  • $n$ is the sample size

Don't worry if this seems a bit complex at first. Practice is key. Keep working through examples, and you'll master these concepts in no time! Good luck, and remember, you've got this!