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DG5 --- 20 minutes

Accel Math IIIUnit #1: Data Analysis

Lesson #8: Normal Distribution

EQ: What are the characteristics of a normal distribution and how is probability calculated using this type of distribution?

Recall: Three Types of Distributions

Binomial Normal Geometric

Normal Distributions --- created from continuous random variables

Characteristics of a Normal Distribution:1. Symmetric, Bell-Shaped Curve and Uni-modal.

2. Mean, Median, Mode are equal and located at the middle of the distribution. Symmetric about the mean. Not skew.

3. The curve is continuous, no gaps or holes. The curve never touches or crosses the x-axis.4.The total area under the curve equals

1.Recall: Empirical Rule

Normal Distribution --- each has its own mean and standard deviation.

What are µ and σ in this normal distribution ?50 10

Standard Normal Distribution --- mean is always 0 and standard deviation is always 1

STANDARDIZE

z-score --- the number of standard deviations above or below the mean

z = observed – mean or z = X - µ standard deviation σ

Correlates to areaunder the curve.

Ex. In a study of bone brittleness, the ages of people at the onset of osteoporosis followed a normal distribution with a mean age of 71 and a standard deviation of 2.8 years. What z-score would an age of 65 represent in this study?

62.6 65.4 68.2 71 73.8 76.6 79.4

| | | | | || | | | | |-3 -2 -1 0 1 2 3

14.28.27165

z

| 65

| -2.14

Using Table A to Finding the Area under A Standard Normal Curve

Ex. Find the area under the curve to the left of z = -2.18.

| -2.18

%46.10146.0)18.2( zP

Ex. Find the area under the curve to the left of z = 1.35.

| 1.35

%15.919115.0)35.1( zP

Ex. Find the area under the curve to the right of z = 0.75.

| .75

%66.222266.07734.01)75.0( zP

WHY??

Ex Find the area under the curve between z = -1.36 and z = 0.42.

Ex. Find the area under the curve between z = 1.60 and z = 3.3.

In Class Practice Worksheet: Area Under the Standard

Normal Curve#1 - 11

What about finding a z-score when given area under the curve?Ex. Determine the z-score that would

give this area under the curve.

Ex. Determine the z-score that would give this area under the curve.

Ex. Determine the z-score that would give this area under the curve.

In Class Practice Worksheet: Area Under the Standard

Normal Curve#12

Practice Worksheet Calculating Area Using z-scores

Quiz 2 --- 30 minutes

Using the Graphing Calculator with Normal Distributions

Command and Arguments: I. When given a z-score, you are

looking for area under the curve.

normalcdf(low bound, high bound)

Ex. Find the area under the curve to the left of z = 1.35.

Ex Find the area under the curve between z = -1.36 and z = 0.42.

Assignment: Example Handout: Area Under

the Standard Normal Curve #1 – 11

Go back and rework these using the calculator function.

II. When given area, you’re looking for a z-score.

Use invnorm function under distributions.invnorm(% to the left)

***Remember invnorm and normcdf are calculator jargon

Do not write these functions on assessments as “work”.Assignment: Redo WS Calculating Area

Using z-scores