Welcome back to Professor Baker's Math Class! In this session, we are diving into Chapter 10: Estimation (Single Samples). Now that we understand the basics of sampling distributions, it's time to put that knowledge to work by estimating population parameters with confidence intervals. Whether you are analyzing the breaking strength of metal chains or predicting election results, these tools are essential for making data-driven decisions.

1. Estimating the Population Mean: The Z-Interval

First, we looked at how to estimate a population mean ($\\mu$) when the population standard deviation ($\\sigma$) is known. This is our "best-case scenario" where we can use the standard Normal Distribution (Z-scores).

The formula for the confidence interval is:

$$ \\bar{x} \\pm z \\left( \\frac{\\sigma}{\\sqrt{n}} \\right) $$

Class Example: In the notes, we reviewed a problem regarding the breaking strength of metal link chains. With a sample size of $n=50$, a known $\\sigma=100$, and a sample mean $\\bar{x}=5000$, we constructed a 99% confidence interval. Since we knew $\\sigma$, we used a Z-score of $2.575$. This allowed us to state with 99% confidence that the true mean strength lies between 4963.58 and 5036.42 pounds.

2. Estimating the Population Mean: The T-Interval

In the real world, we rarely know the population standard deviation. When $\\sigma$ is unknown, we must use the sample standard deviation ($s$) as an estimate. This introduces a bit more uncertainty, so we switch from the Z-distribution to the Student's t-distribution.

Key concepts for T-Intervals:

  • We use Degrees of Freedom ($df$), calculated as $df = n - 1$.
  • As the degrees of freedom increase, the t-distribution looks more like the normal Z-distribution (as seen in Figure 10.2.5 in your notes).
  • The formula changes slightly to: $ \\bar{x} \\pm t \\left( \\frac{s}{\\sqrt{n}} \\right) $

Class Example: We analyzed a manufacturing company measuring production time. Because we only had the sample standard deviation ($s=4.32$) and a small sample size ($n=10$), we used a t-score of $2.262$ (based on $df=9$).

3. Estimating Population Proportions

Finally, we moved away from averages (means) to look at proportions (percentages). This is commonly used in polling, like the voter survey example in our notes.

To find the point estimate (\\hat{p}), we use:

$$ \\hat{p} = \\frac{x}{n} $$

The confidence interval for a proportion always uses a Z-score and the following standard error formula:

$$ \\hat{p} \\pm z \\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}} $$

Class Example: In the Voting, Inc. poll, we found that 220 out of 500 voters favored a candidate (\\hat{p} = 0.44). Using a 95% confidence level ($z=1.96$), we calculated the margin of error to find the true proportion of voters is likely between 39.6% and 48.3%.

Summary Checklist

When solving these problems on your homework or exams, ask yourself:

  1. Is the data quantitative (numbers) or categorical (proportions)? If proportions, use Z-scores and the $\\hat{p}$ formula.
  2. If quantitative (Mean), do I know $\\sigma$?
    • Yes, I know $\\sigma$ $\rightarrow$ Use Z-score.
    • No, I only have $s$ $\rightarrow$ Use t-score.

Keep practicing those calculator functions to find the mean and standard deviation from raw data sets (like the funeral home costs example), and you will do great! See you in the next class.