Section 8.4-8.6: Exploring Normal Distributions and Z-Scores

Hello everyone! In our recent class on October 26th, 2023, we delved into the fascinating world of normal distributions and z-scores, covering sections 8.4 through 8.6. Let's recap the key concepts and examples we discussed to solidify your understanding.

Standardizing Normal Random Variables

A cornerstone of our discussion was how to transform any normal random variable into a standard normal random variable. This is achieved using the following formula, which allows us to use the standard normal distribution table (z-table) to find probabilities:

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

Where:

  • z is the standard normal variable (z-score)
  • x is the normal random variable
  • $\mu$ is the mean of the distribution
  • $\sigma$ is the standard deviation of the distribution

Remember, the z-score tells us how many standard deviations away from the mean a particular value is. A positive z-score indicates the value is above the mean, while a negative z-score indicates it's below the mean.

Calculating Probabilities with Z-Tables

We practiced using standard normal tables (z-tables) to determine probabilities associated with different z-scores. This involves finding the area under the standard normal curve to the left of a given z-score. Here are a few examples:

  1. Finding P(0 ≤ z ≤ 0.79): This requires looking up the z-score of 0.79 in the z-table. The value represents the area under the curve between z = 0 and z = 0.79. The approximate value from the provided document is 0.28524.
  2. Finding P(-1.57 ≤ z ≤ 2.33): Here, we need to look up both z-scores in the table. Find the area to the left of z = 2.33 and subtract the area to the left of z = -1.57 to find the area between them. According to the notes, this is approximately 0.99010 - 0.05821 = 0.93189.
  3. Finding P(z ≥ 1.89): Since the z-table gives the area to the left of the z-score, we calculate this probability as 1 - P(z < 1.89). Using the notes, this is 1-0.97062 = 0.02938.
  4. Finding P(z ≤ -2.77): This is a direct lookup in the z-table. We look for -2.77 and the given probability is about 0.00280.

Applying the Concepts: Real-World Examples

We also looked at applying these concepts to real-world problems. For example:

  • Finding the probability that a normal random variable (e.g., a test score) falls between two values by standardizing those values into z-scores and using the z-table.
  • Calculating what score is needed to fall into a certain percentile (e.g., the 90th percentile)

Remember to practice these calculations and visualizations to solidify your understanding of normal distributions and z-scores. Keep up the great work, and don't hesitate to ask questions if anything is unclear!