Lecture 6: Chapter 6: Normal Probability normal ...

Lecture 6: Chapter 6: Normal Probability Distributions

A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve. A normal distribution has the following properties.

1. The mean, median, and mode are equal.

2. The normal curve is bell shaped and is symmetric about the mean.

3. The total are under the normal curve is equal to one.

4. The normal curve approaches, but never touches, the x-axis as it extends

farther and farther away from the mean.

5. Between and (in the center of curve) the graph curves

downward. The graph curves upward to the left of and to the right of

The points at which the curve changes from curving upward to

curving downward are called inflection points.

6. The Empirical Rule: Approximately 68% of the area under the normal curve

is between and . Approximately 95% of the area under the

normal curve is between

and

. Approximately 99.7% of the

area under the normal curve is between

and

.

6-2 The standard Normal Distribution

Def: The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.

Finding Area under the Standard Normal curve using Table A-2:

When using table A-2, it is essential to understand the following points.

I) Table A-2 is designed only for the standard normal distribution, which has a

mean of 0 and a standard deviation of 1.

II) Each value in the body of the table is the cumulative area from the left op

to a vertical line above a specific value of z. Recall: the z-score allows us

to transform a random variable x with mean and standard deviation

into a random variable z.

with mean 0 and standard deviation of 1.

III) When this transformation takes place, the area that falls in the interval under the nonstandard normal curve is the same as that under the standard normal curve with the corresponding z-boundaries.

IV) Table A-2 is on two pages. One for positive z scores and one for negative z scores.

Example 1: Find the indicated area under the standard normal curve.

a) Between z=0 and z = 1.96 b) To the right of z = 1.64 c) To the left of z = 1.54 d) To the right of z = -0.95 e) To the left of z =-2.57 f) Between z=-0.44 and z = 1.66 g) Between z =1.66 and z = 2.97

Notation:

P(aa) = 1 - p(z ................
................

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