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Normal Probability Distributions

In this section we deal with the the idea of a continuous random variable. By this it is meant that the the random variable can take on an infinite number of values, with the property that there are no gaps between the values. An example of a continuous random variable is the selection of time of the day or the speed of a car. The type of distribution that will be discussed here is the normal distribution.

Introduction to Normal Distributions

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Definition of a normal distribution

A random variable is said to have a normal distribution if it has a probability distribution that is symmetric and bell-shaped see figure 4.1.

Figure 4.1: A Normal Distribution

There two VERY important things to mention here. First, the total area under the curve is 1. The second is area will be used to measure probabilities. A normal distribution is intimately connected to Z-scores. The main idea is to standardize all the data that is given by using Z-scores. These Z-scores can then be used to find the area (and thus the probability) under the normal curve. Before getting into computing probabilities, here is a quick reminder of Z-scores.

Definition of Z-score

A Z-score is the number of standard deviations that a given x value is above or below the mean. If z represents the Z-score for a given x value then

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

(Round your answer to 2 decimal places.)

Using Z-scores to Find Probabilities

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This section deals with converting values to Z-scores so that a standard normal curve can be used to determine the probabilities.

Men's height

The heights of men have a bell-shaped distribution with a mean of 69.0 inches and a standard deviation of 2.8 inches. What percentage of men have heights between 65.4 inches and 72.3 inches? Solution: The Z-score associated with 65.4 is (65.4 - 69.0)/2.8 = -1.29. For 72.3, the Z-score is 1.18 (why?) Since the respective Z-scores are -1.29 and 1.18, the area under the bell-shaped curve is 78.25%. Check it out for yourself! Drag the points below.

The idea of Z-scores is very valuble for understanding what you are calculating when finding probabilities. However, using the TI-83 avoids the explicit calculations of the Z-score. Knowing how to compute Z-scores and find the areas will be VERY useful for later computations.

How to find probabilities using the TI-83

1. Press 2nd VARS.
2. Select normalcdf(.
3. Complete the entry to obtain normalcdf(lower bound, upper bound, Mean, Standard Deviation), substituting the appropriate values in.

Men's height cont.

Do the problem above using the TI-83. Solution: Enter in normalcdf(65.4, 72.3, 69, 2.8). This yields 78.14%.

Note there is a discrepancy between the two answers. The method that involved Z-scores had round off error, while the TI-83 method does not.

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