Day 10 AGENDA: Quiz 2--- 30 minutes

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Lesson #8: Normal Distribution Accel Precalculus 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.

The total area under the curve equals 1. 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. New Term: Empirical Rule 68 _____% of data within 1 standard deviation _____% of data within 2 standard deviations _____% of data within 3 standard deviations 95 99

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

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

Area Under the Standard Normal Curve #1 - 12 Practice Worksheet: Area Under the Standard Normal Curve #1 - 12 Practice Worksheet Calculating Area Using z-scores

Day 11 Agenda:

Handout: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%

Handout: Finding the Area under the Curve 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 your “work”. Must include probability statements.

WS: Calculating Area Using z-scores Assignment: Go back and rework the worksheets using the calculator functions. Example Handout: Area Under the Standard Normal Curve #1 – 12 WS: Calculating Area Using z-scores NOTE: All normal probability problems should include a probability statement and a shaded/labeled SND curve.