The normal distribution

Summary

The normal distribution is used to model a continuous random variable and is represented by a bell-shaped curve. It is defined by its mean (μ) and standard deviation (σ), written as X ~ N(μ, σ²).

• Normal Distribution — a probability distribution for a continuous random variable that is symmetric about the mean. Example: Heights of people in a population.
• Standard Normal Distribution — a normal distribution with a mean of 0 and a standard deviation of 1. Example: Z ~ N(0,1) is used for standardizing values.
• Standardization — transforming a normal distribution to a standard normal distribution by subtracting the mean and dividing by the standard deviation. Example: Converting X ~ N(μ, σ²) to Z ~ N(0,1).
• Probability Tables — tables used to find probabilities and z-values for the standard normal distribution. Example: Finding P(Z < 1.37) using a Z-table.

Exam Tips

Key Definitions to Remember

• Normal Distribution: A continuous probability distribution that is symmetric about the mean.
• Standard Normal Distribution: A normal distribution with mean 0 and standard deviation 1.
• Standardization: The process of converting a normal distribution to a standard normal distribution.

Common Confusions

• Confusing the mean and standard deviation in the normal distribution notation.
• Misinterpreting the Z-table values as probabilities greater than z instead of less than z.

Typical Exam Questions

• What is the probability that a value is greater than a certain number in a normal distribution? Use the Z-table to find the probability.
• How do you find the mean or standard deviation given certain probabilities? Use the properties of the normal distribution and Z-tables.
• How do you use the normal distribution as an approximation to the binomial distribution? Apply continuity correction and use the normal approximation.

What Examiners Usually Test

• Understanding of the properties of the normal distribution.
• Ability to use Z-tables to find probabilities and z-values.
• Application of standardization techniques to solve problems.