Section 6.1 Introduction to the Normal Curve HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.

A continuous probability distribution for a given random variable, X, that is completely defined by it’s mean and standard deviation. HAWKES LEARNING SYSTEMS math courseware specialists Normal Distribution: Continuous Random Variables 6.1 Introduction to the Normal Curve 1.A normal curve is symmetric and bell-shaped. 2.A normal curve is completely defined by its mean, , and standard deviation, . 3.The total area under a normal curve equals 1. 4.The x-axis is a horizontal asymptote for a normal curve. Properties of a Normal Distribution:

HAWKES LEARNING SYSTEMS math courseware specialists Symmetric and Bell-Shaped: Continuous Random Variables 6.1 Introduction to the Normal Curve

HAWKES LEARNING SYSTEMS math courseware specialists Completely Defined by its Mean and Standard Deviation: Continuous Random Variables 6.1 Introduction to the Normal Curve An inflection point is a point on the curve where the curvature of the line changes. The inflection points are located at  and 

HAWKES LEARNING SYSTEMS math courseware specialists Total Area Under the Curve = 1: Continuous Random Variables 6.1 Introduction to the Normal Curve

HAWKES LEARNING SYSTEMS math courseware specialists The x-Axis is a Horizontal Asymptote: Continuous Random Variables 6.1 Introduction to the Normal Curve

a.Birth weights of 75 babies. Normal b.Ages of 250 students in 10 th grade. No, this would be uniform c.Heights of 100 adult males. Normal d.Frequency of outcomes from rolling a die. No, because the data is discrete e.Weights of 50 fully grown tigers. Normal Determine if the following is a normal distribution: HAWKES LEARNING SYSTEMS math courseware specialists Continuous Random Variables 6.1 Introduction to the Normal Curve

HAWKES LEARNING SYSTEMS math courseware specialists How Many Normal Curves are there? Continuous Random Variables 6.1 Introduction to the Normal Curve Because there are an infinite number of possibilities for  and , there are an infinite number of normal curves.

A standard normal distribution has the same properties as the normal distribution; in addition, it has a mean of 0 and a standard deviation of 1. HAWKES LEARNING SYSTEMS math courseware specialists Standard Normal Distribution: Continuous Random Variables 6.1 Introduction to the Normal Curve 1.The standard normal curve is symmetric and bell- shaped. 2.It is completely defined by its mean and standard deviation,  0 and  1. 3.The total area under a standard normal curve equals 1. 4.The x-axis is a horizontal asymptote for a standard normal curve. Properties of a Standard Normal Distribution:

HAWKES LEARNING SYSTEMS math courseware specialists Converting to the Standard Normal Curve: Continuous Random Variables 6.1 Introduction to the Normal Curve Standard Score Formula (z-score): When calculating the z-score, round your answers to two decimal places.

Given  40 and  5, indicate the mean, each of the inflections points, and where each given value of x will appear on the curve. Draw a Normal Curve: HAWKES LEARNING SYSTEMS math courseware specialists Solution: x 1 = 33 and x 2 = Continuous Random Variables 6.1 Introduction to the Normal Curve

Given  40 and  5, calculate the standard score for each x value and indicate where each would appear on the standard normal curve. Convert to the Standard Normal Curve: HAWKES LEARNING SYSTEMS math courseware specialists Solution: x 1 = 33 and x 2 = 11 2.2  1.4 Continuous Random Variables 6.1 Introduction to the Normal Curve

Given  48 and  5, convert to a normal curve and indicate where a score of x = 45 would appear on each standard normal curve. Convert to the Standard Normal Curve: HAWKES LEARNING SYSTEMS math courseware specialists Solution: 11  Continuous Random Variables 6.1 Introduction to the Normal Curve