Copyright © 2014 Pearson Education, Inc. All rights reserved Chapter 6 Modeling Random Events: The Normal and Binomial Models

6 - 2 Copyright © 2014 Pearson Education, Inc. All rights reserved Learning Objectives  Be able to distinguish between discrete and continuous-valued variables.  Know when a Normal model is appropriate and be able to apply the model to find probabilities.  Know when the binomial model is appropriate and be able to apply the model to find probabilities.

Copyright © 2014 Pearson Education, Inc. All rights reserved 6.1 Probability Distributions Are Models of Random Experiments

6 - 4 Copyright © 2014 Pearson Education, Inc. All rights reserved Probability Models and Distributions  A Probability Model is a description of how a statistician thinks data are produced.  A Probability Distribution or Probability Distribution Function (pdf) is a table graph or formula that gives all the outcomes of an experiment and their probabilities.

6 - 5 Copyright © 2014 Pearson Education, Inc. All rights reserved Discrete vs. Continuous  A random variable is called Discrete if the outcomes are values that can be listed or counted.  Number of classes taken  The roll of a die  A random variable is called Continuous if the outcomes cannot be listed because they occur over a range.  Time to finish the exam  Exact weight

6 - 6 Copyright © 2014 Pearson Education, Inc. All rights reserved Discrete or Continuous  Classify the following as discrete or continuous:  Length of the left thumb  Number of children in a the family  Number of devices in the house that connect to the Internet  Sodium concentration in the bloodstream →Continuous →Discrete

6 - 7 Copyright © 2014 Pearson Education, Inc. All rights reserved Discrete Probability Distributions  The most common way to display a pdf for discrete data is with a table.  The probability distribution table always has two columns (or rows).  The first, x, displays all the possible outcomes  The second, P(x), displays the probabilities for these outcomes.

6 - 8 Copyright © 2014 Pearson Education, Inc. All rights reserved Examples of Probability Distribution Tables xP(x) 11/ Die Roll xP(x) Raffle Prize The sum of all the probabilities must equal 1.

6 - 9 Copyright © 2014 Pearson Education, Inc. All rights reserved Examples of Probability Distribution Graphs

Copyright © 2014 Pearson Education, Inc. All rights reserved Continuous Data and Probability Distribution Functions  Often represented as a curve.  The area under the curve between two values of x represents the probability of x being between these two values.  The total area under the curve must equal 1.  The curve cannot lie below the x-axis.

Copyright © 2014 Pearson Education, Inc. All rights reserved 6.2 The Normal Model

Copyright © 2014 Pearson Education, Inc. All rights reserved The Normal Model  The Normal Model is a good model if:  The distribution is unimodal.  The distribution is approximately symmetric.  The distribution is approximately bell shaped.  The Normal Distribution is also called Gaussian.

Copyright © 2014 Pearson Education, Inc. All rights reserved Center and Spread of the Normal Distribution   stands for the center or mean of a distribution.   stands for the standard deviation of a distribution  Note that the Greek letters  and  are used for distributions and x and s are used for sample data.

Copyright © 2014 Pearson Education, Inc. All rights reserved Notation and Area  N(6,2) means the normal distribution with mean  = 6 and standard deviation  = 2.  The area under the normal curve, above the x-axis, and to the left of x = 4 represents P(x < 4).  P(x < 4) = P(x ≤ 4) for a continuous variable.

Copyright © 2014 Pearson Education, Inc. All rights reserved Example: Baby Seals  Research has shown that the mean length of a newborn Pacific harbor seal is 29.5 in. and that  = 1.2 in. Suppose that the lengths follow the Normal model. Find the probability that a randomly selected pup will be more than 32 in.  P(x > 32) ≈ 0.019

Copyright © 2014 Pearson Education, Inc. All rights reserved The Normal Model and the Empirical Rule  The Empirical Rule told us that if a distribution is approximately normal, then 68% of the data will fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.  If the distribution is exactly normal, then these numbers are just the corresponding areas under the normal curve.

Copyright © 2014 Pearson Education, Inc. All rights reserved 6.3 The Binomial Model

Copyright © 2014 Pearson Education, Inc. All rights reserved The Binomial Model  The Binomial Model applies if: 1. There are a fixed number of trials. 2. Only two outcomes are possible for each trial: Yes or No, Success or Failure, Heads or Tails, etc. 3. The probability of success, p, is the same for each trial. 4. The trials are independent.

Copyright © 2014 Pearson Education, Inc. All rights reserved Binomial or Not?  40 randomly selected college students were asked if they selected their major in order to get a good job.  Binomial  35 randomly selected Americans were asked what country their mothers were born.  Not Binomial, more than two possible answers per trial.  To estimate the probability that students will pass an exam, the professor records a study group’s success on the exam.  Not Binomial, since the outcomes are not independent.

Copyright © 2014 Pearson Education, Inc. All rights reserved Words and Inequalities  Exactly  Less Than  At Least  More Than  At Most → = → < → => → > → <=  Notice that “Less Than” and “At Least” are complements and “More Than” and “At Most” are Complements.