Section 6.1

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Section 6.1

Introduction to the Normal Distribution

With extra good stuff added by D.R.S., University of Cordele.






Properties of a Normal Distribution

Properties of a Normal Distribution 1. A normal distribution is bell-shaped and symmetric

about its mean. 2. A normal distribution is completely defined by its

mean, m, and standard deviation, s. 3. The total area under a normal distribution curve

equals 1. 4. The x-axis is a horizontal asymptote for a normal

distribution curve.






The Standard Normal Distribution

Properties of the Standard Normal Distribution 1. The standard normal distribution is bell-shaped and

symmetric about its mean. 2. The standard normal distribution is completely

defined by its mean, m = 0, and standard deviation, s = 1.

3. The total area under the standard normal distribution curve equals 1.

4. The x-axis is a horizontal asymptote for the standard normal distribution curve.

The two things that make it “The Standard…”






Example 6.1: Calculating and Graphing z-Values

Given a normal distribution with μ = 48 and s = 5, convert an x-value of 45 to a z-value and indicate where this z-value would be on the standard normal distribution. Solution Begin by finding the z-score for x = 45 as follows.

45 485

0.60x

z

TI-84 Needs Extra Parentheses!!! ( 4 5 ─ 4 8 ) / 5






Example 6.1: Calculating and Graphing z-Values (cont.)

Now draw each of the distributions, marking a standard score of z = −0.60 on the standard normal distribution.






Example 6.1: Calculating and Graphing z-Values (cont.)

The distribution on the left is a normal distribution with a mean of 48 and a standard deviation of 5. The distribution on the right is a standard normal distribution with a standard score of z = −0.60 indicated.

Recommend two parallel axes, z and x both

RECOMMENDED:1. Sketch a normal

distribution



distribution2. A z-axis, labeling

z = 0 and z = +1, -1, +2, -2, +3, -3, etc., as needed.

For this particular problem, z = -1 and z = +1 are good to visualize,and z = -0.60 is meaningful.




z = 0 and z = +1, -1, +2, -2, +3, -3, as needed.

3. An x-axis, too, labeling the mean and other key values

4. Observe the correspondence between z values and x values.

Corresponding to those fourz-values are these four x-values.




z = 0 and z = +1, -1, +2, -2, +3, -3, as needed.

3. An x-axis, too, labeling the mean and other key values. Observe the correspondence between z values and x values.

It’s clear to the reader that the mean is x = 48.

It’s clear to the reader that for x=45, z=-0.60.

Excel: STANDARDIZE(x value, mean, standard deviation)

Going the other way: converting z back to an x

If you take this formula:

And solve to get x by itself on one side:

45 485

0.60x

z

Section 6.1

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Transcript of Section 6.1