3-21-2023 Class: Sections 6.1 & 6.2

Hi everyone,

I sincerely apologize for forgetting to record our class on March 21st. I understand how important these recordings are for review and catching up. To help make up for this, I've attached the class notes from last semester which cover sections 6.1 and 6.2. Please be aware that this resource also goes into subsequent sections, but the focus will be on 6.1 and 6.2 for this week. Let's get started!

Key Concepts: Density Curves (Section 6.1)

Let's review some fundamental properties of density curves:

  • Property 1: A density curve is always on or above the horizontal axis. This means the y-values are always non-negative.
  • Property 2: The total area under a density curve (and above the horizontal axis) equals 1. This represents the total probability.

These properties are vital because they ensure that a density curve can be used to model probabilities. For example, consider a variable that represents the crystal size of a mineral. Suppose the equation of its density curve is given by $y = \frac{x}{2}$ for $0 < x < 2$. The area under the curve to the left of a certain value $x$ (between 0 and 2) represents the probability that the crystal size is less than $x$. This area is calculated as $\frac{x^2}{4}$. So if $x = 1$, the area is $\frac{1^2}{4} = 0.25$, or 25%. This means there is a 25% chance that a crystal is less than 1 mm.

Key Concepts: Normal Distribution (Section 6.2)

A variable is said to be a normally distributed variable if its distribution has the shape of a normal curve (bell curve). The normal distribution is described by two parameters:

  • $\mu$ (mu): The mean, which represents the center of the distribution.
  • $\sigma$ (sigma): The standard deviation, which measures the spread or variability of the distribution.

The standard normal distribution is a special case where the mean is 0 and the standard deviation is 1.

To work with normal distributions that aren't standard normal, we use the z-score. The z-score is calculated as:

$$z = \frac{x - \mu}{\sigma}$$

Where:

  • $x$ is the value we're interested in.
  • $\mu$ is the mean of the distribution.
  • $\sigma$ is the standard deviation of the distribution.

The z-score tells us how many standard deviations away from the mean a particular value is. Using the z-score allows us to use the Standard Normal Table to determine the probability (area under the standard normal curve) associated with that value.

Basic 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.

Using the Standard Normal Table

The attached notes provide examples of how to use the Standard Normal Table (also called the z-table) to find areas under the standard normal curve. Key things to remember:

  • To find the area to the left of a z-score, simply look up the z-score in the table.
  • To find the area to the right of a z-score, subtract the table value from 1.
  • To find the area between two z-scores, find the table values for both z-scores and subtract the smaller value from the larger value.

For instance, to find the area to the left of $z = 1.23$, you'd look up 1.23 in the table, which gives you 0.8907. This means 89.07% of the data falls to the left of $z = 1.23$.

Please review these notes carefully and don't hesitate to ask questions during our next Q&A session or office hours. We'll work through examples together to solidify your understanding. Keep up the great work!