1. Normal Probability Distributions6
Normal Probability Distributions
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2. Copyright © Cengage Learning. All rights reserved.
6.1
Normal Probability
Distributions
Copyright © Cengage Learning. All rights reserved.

3. Normal Probability Distributions
The normal probability distribution is considered the single most important probability distribution. An unlimited number of continuous random variables have
Either a normal or an approximately normal
Distribution.
Several other probability distributions of both discrete and continuous random variables are also approximately normal under certain conditions.
The normal probability distribution has a continuous random variable and uses two functions: one function to determine the ordinates (y values) of the graph displaying the distribution and a second to determine the probabilities.

4. Normal Probability Distributions
Formula (6.1) expresses the ordinate (y value) that corresponds to each abscissa (x value).
Normal Probability Distribution Function
(6.1)

5. Normal Probability Distributions
When a graph of all such points is drawn, the normal (bell-shaped) curve will appear as shown in Figure 6.1
The Normal Probability Distribution
Figure 6.1

6. Normal Probability Distributions
Note Each different pair of values for the mean, μ, and standard deviation, σ, will result in a different normal probability distribution function.
Formula (6.2) yields the probability associated with the interval from x = a to x = b:
(6.2)

7. Normal Probability Distributions
The probability that x is within the interval from x = a to x = b is shown as the shaded area in Figure 6.2.
Shaded Area: P (a ≤ x ≤ b)
Figure 6.2

8. Normal Probability Distributions
The definite integral of formula (6.2) is a calculus topic and is mathematically more advanced than what is expected in elementary statistics.
Instead of using formulas (6.1) and (6.2), we will use a table to find probabilities for normal distributions.
Formulas (6.1) and (6.2) were used to generate that table.
Before we learn to use the table, however, it must be pointed out that the table is expressed in “standardized” form.

9. Normal Probability Distributions
It is standardized so that this one table can be used to find probabilities for all combinations of mean, μ, and standard deviation , σ, values. That is, the normal probability distribution with mean 38 and standard deviation 7 is similar to the normal probability distribution with mean 123 and standard deviation 32.
As we know that, the empirical rule and the percentages of the distribution that fall within certain intervals of the mean.
The same three percentages hold true for all normal distributions.