Probability and the Normal Curve Chapter 5 This multimedia product and its contents are protected under copyright law. The following are prohibited by law: Any public performance or display, including any transmission of any image over a network; Preparation of any derivative work, including the extraction, in whole or in part, of any images; Any rental, lease, or lending of the program. Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Introduction to Part II In Part I, we learned to categorize data to see basic patterns and trends. Measures of central tendency and variability allowed more descriptions. In Part II, we’ll move towards using statistics to assist us in decision making. Using the concept of probability Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Copyright © Pearson Education, Inc., Allyn & Bacon 2009 Rules of Probability Probability: the relative likelihood of occurrence of any given outcome or event Probability Rules of probability: Converse Rule Addition Rule Multiplication Rule Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Copyright © Pearson Education, Inc., Allyn & Bacon 2009 Converse Rule Converse rule: determines the probability that something will not occur Subtract the probability that something will occur from 1 to find the probability that the event will not occur. Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Copyright © Pearson Education, Inc., Allyn & Bacon 2009 Addition Rule Addition rule: determine cumulative probability by adding probabilities. The probability of obtaining any one of several different and distinct outcomes equals the sum of their separate probabilities. Outcomes must be mutually exclusive. Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Copyright © Pearson Education, Inc., Allyn & Bacon 2009 Multiplication Rule Sometimes we want to know the probability of successive outcomes. Multiplication rule: combination of independent outcomes equals the product of their separate probabilities Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Probability Distributions Probability distributions analogous to frequency distributions Probability distributions based on probability theory Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Copyright © Pearson Education, Inc., Allyn & Bacon 2009 Probability Example Let’s flip a coin two times. probability of heads the first time is .5 probability of heads on the second flip is .5 Using the multiplication rule, probability of getting heads on both flips is .25 Similarly, probability of getting no heads on two flips is .25. Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Probability Example (cont.) What is the probability that one flip in two flips will land on heads? When you add the probabilities of all of the options, they must equal 1. Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Mean of a Probability Distribution Probability distribution has a mean – known as an expected value. Greek letter mu used to indicate the mean in a probability distribution Don’t confuse mu with X bar in a frequency distribution). The Greek letter mu is shown as: μ Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Standard Deviation of a Probability Distribution Probability distribution also has a standard deviation Symbolized by the Greek letter sigma, σ Variance in probability distributions is shown by σ2 s2 for observed data; σ2 for theoretical distribution Copyright © Pearson Education, Inc., Allyn & Bacon 2009

The Normal Curve as a Probability Distribution The normal curve is perfectly symmetrical. Precepts of the normal curve allow for a number of assumptions based on probability theory. But, what are the properties of the normal curve? Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Characteristics of the Normal Curve Sometimes referred to as a bell-shaped curve – perfectly symmetrical. Normal curve is unimodal. Mean, median, mode coincide. Both continue infinitely ever closer but without touching the baseline. The normal curve is a theoretical ideal because it is a probability distribution. Copyright © Pearson Education, Inc., Allyn & Bacon 2009

The Model and the Reality of the Normal Curve The normal curve is a theoretical ideal. Some variables do not conform to the normal curve. Many distributions are skewed, multi-modal, and symmetrical but not bell-shaped. Consider wealth – more “haves” than “have nots” Often, radical departures from normality Copyright © Pearson Education, Inc., Allyn & Bacon 2009

The Area under the Normal Curve 100% of the cases in a normal distribution fall under normal curve Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Copyright © Pearson Education, Inc., Allyn & Bacon 2009

The Area under the Normal Curve Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Copyright © Pearson Education, Inc., Allyn & Bacon 2009 Using Table A Method exists to determine distance from the mean for standard deviations that are not whole numbers. Table A in Appendix C Sigma distances labeled Values for one side of the normal curve given because of symmetry Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Standard Scores and the Normal Curve It’s possible to determine area under the curve for any sigma difference from the mean. This distance is called a z score or standard score z score – indicates direction and degree that any raw score deviates from the mean in sigma units z scores by obtaining the deviation Where: µ=mean of a distribution σ=standard deviation of a distribution z=standard score Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Finding Probability under the Normal Curve Ideas of the normal curve can be used in conjunction with z scores. Normal cure is a probability distribution. Probabilities under normal curve always out of 100% Use Table A to find the probability Earlier rules (converse, addition and multiplication) still apply Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Finding Scores from Probability Based on the Normal Curve Using the formula: it is possible to calculate score values Copyright © Pearson Education, Inc., Allyn & Bacon 2009

Copyright © Pearson Education, Inc., Allyn & Bacon 2009 Summary Introduction to probability Foundation for decision making in statistics Normal curve is a theoretical ideal Use of the normal curve assists in understanding standard deviation z scores used to determine area between and beyond a given sigma distance from the mean Also able to calculate score values for a given z Copyright © Pearson Education, Inc., Allyn & Bacon 2009