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The Normal Distribution Lecture 20 Section 6.3.1 Robb T. Koether Hampden-Sydney College Wed, Sep 28, 2011 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 1 / 41

Outline 1 Density Functions Uniform Distributions 2 The Normal Distribution Examples 3 The 68-95-99.7 Rule 4 The Standard Normal Distribution Standard Normal Areas TI-83 Standard Normal Areas 5 Areas under Other Normal Curves 6 IQ Scores 7 Assignment 8 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 2 / 41

Density Functions Definition (Density Function) A density function is a function whose graph shows the distribution of a population. 0 1 2 3 4 5 6 7 8 A density function is scaled so that its total area is 1. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 4 / 41

Density Functions Definition (Density Function) A density function is a function whose graph shows the distribution of a population. 0.30 0.20 0.10 Area = 1 0.00 0 1 2 3 4 5 6 7 8 A density function is scaled so that its total area is 1. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 4 / 41

Density Functions AREA = PROPORTION = PROBABILITY This is the fundamental property that connects the graph of a continuous model to the population that it represents, namely: The area under the graph between two points on the x-axis represents the proportion of the population that lies between those two points. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 5 / 41

Density Functions Density 0 1 2 3 4 5 6 7 Consider an arbitrary continuous distribution. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 6 / 41

Density Functions Density 0 1 2 3 4 5 6 7 The area under the curve between 2 and 5 is the proportion of the values of x that lie between 2 and 5. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 6 / 41

Density Functions Density Area = Proportion 0 1 2 3 4 5 6 7 The area under the curve between 2 and 5 is the proportion of the values of x that lie between 2 and 5. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 6 / 41

Uniform Distributions The simplest of all density functions is the uniform density function. The graph is horizontal. We will learn more about uniform distributions later. 0.30 0.20 0.10 0.00 Area = 1 0 1 2 3 4 5 6 7 8 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 8 / 41

The Normal Distribution Definition (Normal distribution) The normal distribution is the statistician s name for the bell curve. It is a density function in the shape of a bell. Symmetric. Unimodal. Extends over the entire real line (no endpoints). Main part lies within ±3σ of the mean. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 10 / 41

The Normal Distribution The curve has a bell shape, with infinitely long tails in both directions. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 11 / 41

The Normal Distribution The mean µ is located in the center, at the peak. µ Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 11 / 41

The Normal Distribution The width of the main part of the curve is 6 standard deviations wide (3 standard deviations each way from the mean). σ µ 3σ µ µ + 3σ Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 11 / 41

The Normal Distribution The area under the entire curve is 1. (The area outside of µ ± 3σ is approx. 0.0027.) Area = 1 µ 3σ µ µ + 3σ Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 11 / 41

The Normal Distribution The normal distribution with mean µ and standard deviation σ is denoted N(µ, σ). For example, if X is a variable whose distribution is normal with mean 30 and standard deviation 5, then we say that X is N(30, 5). Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 12 / 41

The Normal Distribution If X is N(30, 5), then the distribution of X looks like this: 0.08 0.06 0.04 0.02 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 14 / 41

Some Normal Distributions Some other examples: N(3, 1). 0.6 0.4 0.2 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 15 / 41

Some Normal Distributions Some other examples: N(5, 1). 0.6 0.4 0.2 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 15 / 41

Some Normal Distributions Some other examples: N ( 2, 1 2). 0.6 0.4 0.2 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 15 / 41

Some Normal Distributions Some other examples: N ( 3 1 2, 1 1 2). 0.6 0.4 0.2 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 15 / 41

The 68-95-99.7 Rule The 68-95-99.7 Rule For any normal distribution, Approximately 68% of the values lie within one standard deviation of the mean. Approximately 95% of the values lie within two standard deviations of the mean. Approximately 99.7% of the values lie within three standard deviations of the mean. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 17 / 41

The Empirical Rule The Empirical Rule For any mound-shaped distribution, Approximately 68% lie within one standard deviation of the mean. Approximately 95% lie within two standard deviations of the mean. Nearly all lie within three standard deviations of the mean. The well-known Empirical Rule is somewhat more general than the 68-95-99.7 Rule. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 18 / 41

The Standard Normal Distribution Definition (Standard normal distribution) The standard normal distribution is the normal distribution with mean 0 and standard deviation 1. It is denoted by the letter Z. That is, Z is N(0, 1). Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 20 / 41

The Standard Normal Distribution 0.3 0.2 0.1 3 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 21 / 41

Areas Under the Standard Normal Curve Easy questions: What is the total area under the curve? What proportion of values of Z will fall below 0? What proportion of values of Z will fall above 0? Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 23 / 41

Areas Under the Standard Normal Curve What proportion of values will fall below +1? 0.3 0.2 0.1 3 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 24 / 41

Areas Under the Standard Normal Curve It turns out that the area to the left of +1 is 0.8413. 0.3 0.2 0.1 3 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 24 / 41

Areas Under the Standard Normal Curve What is the area to the right of +1? 0.3 0.2 0.1 3 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 24 / 41

Areas Under the Standard Normal Curve What is the area between 1 and 1? 0.3 0.2 0.1 3 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 24 / 41

Areas Under the Standard Normal Curve What is the area between 0 and 1? 0.3 0.2 0.1 3 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 24 / 41

Areas Under the Standard Normal Curve There are two methods to finding standard normal areas: The TI-83 function normalcdf. Standard normal table. We will use the TI-83 (unless you want to use the table). Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 25 / 41

TI-83 - Standard Normal Areas TI-83 Standard Normal Areas Press 2nd DISTR. Select normalcdf (Item #2). Enter the lower and upper bounds of the interval. If the interval is infinite to the left, enter E99 as the lower bound. If the interval is infinite to the right, enter E99 as the upper bound. Press ENTER. The area appears in the display. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 27 / 41

Standard Normal Areas Practice Use the TI-83 to find the following. The area between 1 and 1. The area to the right of 1. The area to the left of 1.645. What standard normal percentile is 1.645? Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 28 / 41

Other Normal Curves If we are working with a different normal distribution, say N(30, 5), then how can we find areas under the curve? Use the same procedure as before, except enter the mean and standard deviation as the 3rd and 4th parameters of the normalcdf function. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 30 / 41

TI-83 - Area Under Normal Curves Example (TI-83 Normal Areas) Find area between 25 and 38 in the distribution N(30, 5). In the TI-83, enter normalcdf(25,38,30,5). Press ENTER. The answer 0.7865 appears. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 31 / 41

IQ Scores Read the article Understanding and Interpreting IQ. IQ scores are normalized to have a mean of 100 and a standard deviation of 15. Psychologists often assume a normal distribution of IQ scores as well. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 33 / 41

IQ Scores Practice What percentage of the population has an IQ above 120? above 140? What percentage of the population has an IQ between 75 and 125? Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 34 / 41

Assignment Homework Read Section 6.1-6.3.1, pages 357-370. Let s Do It! 6.1, 6.2, 6.3, 6.4, 6.5, 6.6. Exercises 1-9, 11, 12, 15, 16, 18,, 32, page 376. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 36 / 41

Answers to Even-numbered Exercises Page 376, Exercises 4, 6, 8, 12, 16, 18 6.4 A score at the mean. 6.6 µ = 5 and σ = 1. The Rule says that about 95% lie between µ 2σ and µ + 2σ, which is the interval from 1 to 9. 6.8 (a) Stats: 0.41; Logic: 0.70; Verbal: 1.91. (b) Stats. (c) 65.82%. (d) 24.15%. (e) 2.83%. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 38 / 41

Answers to Even-numbered Exercises Page 376, Exercises 4, 6, 8, 12, 16, 18 6.12 (a) 0.015 0.010 0.005 (b) 0 0.015 0.010 0.005 0.0548. 0 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 39 / 41

Answers to Even-numbered Exercises Page 376, Exercises 4, 6, 8, 12, 16, 18 6.16 The means are the same, but the standard deviation for men is larger than the standard deviation for women. 0.025 0.020 0.015 0.010 0.005 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 40 / 41

Answers to Even-numbered Exercises Page 376, Exercises 4, 6, 8, 12, 16, 18 6.18 (a) 0.2183. (b) 10.19 minutes. (c) 14.36 minutes. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 41 / 41