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Lesson #8: Normal Distribution Accel Math III Unit #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

1. Symmetric, Bell-Shaped Curve and Uni-modal. 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 3. The curve is continuous, no gaps or holes. The curve never touches or crosses the x-axis. 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 area under the curve. Area under the curve at that score z-score

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? | | | | | | | -2.14 | | | | | | | 65 -3 -2 -1 0 1 2 3 62.6 65.4 68.2 71 73.8 76.6 79.4

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

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

Ex. Find the area under the curve to the right of z = 0.75. We want the area to the RIGHT WHY?? | .75

Ex Find the area under the curve between z = -1.36 and z = 0.42. P(-1.36 < z < 0.42) =_____ 0.6628 – 0.0869 =_____ 0.5759 -1.36 0.42

Ex. Find the area under the curve between z = 1.60 and z = 3.3. P(1.60 < z < 3.3) =_____ 0.0543 0.9995 – 0.9452 =_____ 1.60 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. -0.52

Ex. Determine the z-score that would give this area under the curve. 0.67

Ex. Determine the z-score that would give this area under the curve. -1.13

In Class Practice Worksheet: Area Under the Standard Normal Curve #12a – e only Practice Worksheet Calculating Area Using z-scores

Using the Graphing Calculator with Normal Distributions Command and Arguments:   When given a z-score, you are looking for area under the curve. normalcdf(low bound, high bound) normcdf(___, ____) -10 -2.18 P(z <-2.18) = _____ 0.0146 = 1.46%

normcdf(___, ____) -10 1.35 P(z < 1.35) = _____ 0.9115 = 91.15% Ex. Find the area under the curve to the left of z = 1.35. normcdf(___, ____) -10 1.35 P(z < 1.35) = _____ 0.9115 = 91.15% normcdf(___, ____) 0.75 10 P(z > 0.75) = _____ 0.2266 = 22.66%

normcdf(___, ____) P(-1.36 < z < 0.42) = _____ 0.5758 = 57.58% Ex Find the area under the curve between z = -1.36 and z = 0.42. normcdf(___, ____) -1.36 0.42 P(-1.36 < z < 0.42) = _____ 0.5758 = 57.58% normcdf(___, ____) 1.60 3.3 P(1.60 < z < 3.3) = _____ 0.0543 = 5.43%

When given area, you’re looking for a z-score. Use invnorm function under distributions. invnorm(% to the left)

1.64 0.95 0.67 0.25 0.75 Pay attention to what you write!

1.20 0.885 0.50 -1.15 1.15 0.125 0.375 0.375 0.125 or 0.875 Symmetric 0.50

***Remember invnorm and normcdf are calculator jargon Do not write these functions on assessments as “work”. Must include probability statements.

Example Handout: Area Under the Standard Normal Curve #1 – 12 Assignment: Example Handout: Area Under the Standard Normal Curve #1 – 12 WS Calculating Area Using z-scores Go back and rework these using the calculator function.