Overview
This lecture introduces practical applications of the normal distribution, explaining how to solve problems involving real-world data modeled by the normal curve.
Introduction to the Normal Distribution
- The normal distribution is a symmetric, bell-shaped curve used to model many natural phenomena.
- It is commonly applied to real-world data such as heights, weights, and test scores.
- Standard normal distribution has a mean (μ) of 0 and a standard deviation (σ) of 1.
Applying the Normal Distribution
- To solve problems, raw data is standardized using the z-score formula: ( z = \frac{(x - \mu)}{\sigma} ).
- The z-score represents how many standard deviations a value (x) is from the mean.
- Areas under the normal curve correspond to probabilities or percentages of occurrence.
- Problems often ask for the probability of a value being above, below, or between certain points.
Using Z-tables
- Z-tables provide the area (probability) to the left of a given z-score under the standard normal curve.
- To find the probability for a value range, convert endpoints to z-scores and use the table values.
Key Terms & Definitions
- Normal Distribution — a symmetric, bell-shaped statistical distribution defined by its mean and standard deviation.
- Mean (μ) — the average or central value.
- Standard Deviation (σ) — a measure of spread or variability.
- Z-score — the number of standard deviations a value is from the mean.
- Z-table — a table showing cumulative probabilities for standard normal z-scores.
Action Items / Next Steps
- Practice converting data values to z-scores using the formula.
- Use z-tables to find probabilities for given scenarios.
- Complete any assigned homework or reading on normal distribution applications.