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Chapter 6: Normal Probability Distributions

6-1 Overview

Continuous random variables do not have gaps between the numbers. For example, height can be 1, 1.1, 1.01, 1.001 or whatever precision you need. Thus graphs of the probability distributions are smooth and do not have the vertical bars that are required by graphs of discrete probability distributions.

6-2 The Standard Normal Distribution

All Normal Probability Distributions have a bell or mound shape. The Standard Normal Distribution is the normal distribution with μ = 0 and σ = 1. Remember μ is the mean and σ is the standard deviation of a population. Use Table A-2 or STATDISK to look up the probability associated with z.

6-3 Applications of Normal Distributions

The z-score formula for n = 1 is: $z=\frac{x-\mu }{\sigma }$

Any normal probability distribution can be transformed into a standard normal probability distribution using the z formula, so that you can use Table A-2 to look up probabilities to solve problems.

6-4 Sampling Distributions & Estimators

Because we seldom have access to the entire population, we need to develop the relationship between the population parameters (what we want to know but can directly compute) and the sample statistics (what we can compute but know are uncertain). That relationship is called the Central Limit Theorem.

6-5 The Central Limit Theorem

The z-score formula for n > 1 is $z=\frac{x-\mu }{\frac{\sigma }{\sqrt{n}}}$

In spite of the mathematical complexity behind the development of The Central Limit Theorem, it is very easy to apply it to our work. The z formula changes to incorporate samples larger than 1 and we use the same problem solving techniques learned in 6-3

6-6 Normal as Approximation to Binomial

Surprisingly, many discrete binomial probability distributions look approximately like continuous normal probability distributions. In those cases we can expand our application of the properties of normal distributions to binomial distributions.