Class Notes: Chapter 8 Part 1 - Introduction to Confidence Intervals

Welcome to Chapter 8! Today, we're diving into the world of estimation. Specifically, we'll be focusing on estimating population means using sample data. Get ready to learn about point estimates and confidence intervals, powerful tools for making informed decisions based on data.

Point Estimates

Let's start with the basics. A point estimate is a single value that best approximates a population parameter. For example, if we want to estimate the average price, $\mu$, of all new mobile homes, we can use the sample mean, $\bar{x}$, calculated from a sample of new mobile homes.

Consider a sample of 36 new mobile homes (see Table 8.1 in the attached notes). We can calculate the sample mean price as follows:

$$ \bar{x} = \frac{\Sigma x_i}{n} = \frac{2278}{36} = 63.28 $$

This means our point estimate for the average price of all new mobile homes is $63.28 thousand (or $63,280). Remember, a point estimate is simply the value of a statistic used to estimate the parameter.

Confidence Intervals

While point estimates are useful, they don't tell us how accurate our estimate is. That's where confidence intervals come in! A confidence interval provides a range of values within which we believe the true population parameter lies, with a certain level of confidence.

A confidence interval (CI) is an interval of numbers obtained from a point estimate of a parameter.

The confidence level indicates how confident we are that the parameter lies within the confidence interval. For example, a 95% confidence level means that if we were to repeat the sampling process many times, 95% of the resulting confidence intervals would contain the true population mean, $\mu$.

The combination of the confidence level and confidence interval is called the confidence-interval estimate.

Calculating a Confidence Interval (One-Mean z-Interval Procedure)

Here's how to construct a confidence interval for a population mean when the population standard deviation, $\sigma$, is known:

  1. Assumptions:
    • We have a simple random sample.
    • The population is normally distributed or the sample size is large enough (Central Limit Theorem).
    • The population standard deviation, $\sigma$, is known.
  2. Formula: The confidence interval for $\mu$ is given by: $$\bar{x} - z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} \text{ to } \bar{x} + z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$ Where:
    • $\bar{x}$ is the sample mean.
    • $z_{\alpha/2}$ is the z-score corresponding to the desired confidence level (e.g., for a 95% confidence level, $z_{\alpha/2} = 1.96$).
    • $\sigma$ is the population standard deviation.
    • $n$ is the sample size.
  3. Interpretation: Interpret the confidence interval. For example, "We are 95% confident that the true population mean lies between [lower bound] and [upper bound]."

Example: Mobile Home Prices

Let's go back to our mobile home example. Suppose the population standard deviation for the price of new mobile homes is known to be $\sigma = $7.2 thousand. We want to find a 95% confidence interval for the population mean price, $\mu$.

We already know $\bar{x} = 63.28$ and $n = 36$. For a 95% confidence level, $z_{\alpha/2} = 1.96$. Therefore, $$\frac{\sigma}{\sqrt{n}} = \frac{7.2}{\sqrt{36}} = 1.2$$

The 95% confidence interval is:

$$63.28 \pm (1.96)(1.2) = 63.28 \pm 2.352$$

This gives us an interval of approximately $60.93$ to $65.63$ (in thousands of dollars). So, we can be 95% confident that the true average price of all new mobile homes lies between $60,928 and $65,632.

Remember, understanding confidence intervals allows us to make more informed judgments about population parameters based on sample data. Keep practicing, and you'll master this crucial concept!