Important Update: Online Class on October 31st

Hello Class,

Due to a potential COVID-19 exposure, I held class online via Zoom on October 31st. The recording is available if you couldn't make it. We discussed test grades at the beginning of the class, and if you'd like your score, just email me. Thank you for your understanding!

Here is the zoom link for the class: https://us02web.zoom.us/j/86541211281

Chapter 6: The Normal Distribution

This week, we began exploring the normal distribution, a cornerstone of statistical analysis. Here's a recap of what we covered:

  • 6.1 Introducing Normally Distributed Variables: We started by defining what it means for a variable to be normally distributed. Remember, a normal distribution is often called a "bell curve" due to its shape.
  • 6.2 Areas under the Standard Normal Curve: A key skill is calculating areas under the standard normal curve. These areas represent probabilities. We often use a Z-table to find these areas.
  • 6.3 Working with Normally Distributed Variables: We learned how to standardize normal variables using the Z-score formula: $$Z = \frac{X - \mu}{\sigma}$$, where $X$ is the data point, $\mu$ is the mean, and $\sigma$ is the standard deviation. The Z-score tells us how many standard deviations a data point is from the mean.
  • 6.5 Normal Approximation to the Binomial Distribution (Optional): We briefly touched upon using the normal distribution to approximate binomial probabilities under certain conditions (large $n$). This is a powerful tool when dealing with many trials.

Key Concepts and Properties

Here are some important properties of the standard normal curve:

  • Property 1: The total area under the standard normal curve is 1.
  • Property 2: The standard normal curve extends indefinitely in both directions, approaching but never touching the horizontal axis.
  • Property 3: The standard normal curve is symmetric about 0.
  • Property 4: Almost all the area under the standard normal curve lies between -3 and 3.

Z-Scores Explained

A Z-score represents the number of standard deviations a given data point is away from the mean. For example, if we determine the area under the standard normal curve to the left of $z = 1.23$ is $0.8907$, this gives us the probability of observing a value less than 1.23 standard deviations above the mean.

Empirical Rule

Don't forget the Empirical Rule (68-95-99.7 rule):

  • Approximately 68% of the data falls within one standard deviation of the mean ($\mu \pm \sigma$).
  • Approximately 95% of the data falls within two standard deviations of the mean ($\mu \pm 2\sigma$).
  • Approximately 99.7% of the data falls within three standard deviations of the mean ($\mu \pm 3\sigma$).

Keep practicing, and you'll master the normal distribution in no time!