CHAPTER 4 RANDOM VARIABLES AND DISTRIBUTIONS

1 CHAPTER 4 RANDOM VARIABLES AND DISTRIBUTIONSE very baby is born either male or female. So every child is the outcome of a human binomial 1308/31/11 6:49 PM8/31/11 6:49 PMCopyright 2011, K12 Inc. All rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the express prior written consent of K12 2011, K12 Inc. All rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the express prior written consent of K12 This ChapterYou will learn how RANDOM VARIABLES and probability DISTRIBUTIONS can describe problem situations. You ll also learn how to use expected value to make decisions. Specific discrete and continuous DISTRIBUTIONS , such as binomial and normal DISTRIBUTIONS , will be covered as well as techniques for using the standard normal distribution to compare List CHAPTER 4 Introduction Creating Probability DISTRIBUTIONS Interpreting Probability DISTRIBUTIONS Expected Value Binomial DISTRIBUTIONS Continuous RANDOM VARIABLES The Normal distribution Standardizing Data Comparing Scores The Standard Normal Curve Finding Standard Scores CHAPTER 4 Wrap-UpIn a family with one child, what is the chance that the child is a girl?

2 In a family with two children, what is the chance that both of them are girls? What about the chance of having three girls, or four, or five? RANDOM VARIABLES can help describe situations like VARIABLES AND DISTRIBUTIONS 1318/31/11 6:49 PM8/31/11 6:49 PMCopyright 2011, K12 Inc. All rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the express prior written consent of K12 2011, K12 Inc. All rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the express prior written consent of K12 individual events can be RANDOM , they can yield patterns when repeated multiple DistributionsIt is often predicted that the probability that a newborn baby will be a boy (or girl, as the case may be) is 50%.

3 Due to the natural gender ratio, however, the actual probability that a newborn will be a boy is not exactly 50%. In the United States, for example, the chances are slightly higher that a newborn will be a boy rather than a relative frequency histogram shows the proportions for the number of boys in a family with 3 children, given the predicted probability of 50%.According to this graph, there is a chance of having exactly 1 boy out of 3 children. You can also see that there is a 25% chance that all 3 children will have the same gender (just add the heights of the first and last bar).The graph for the actual number of boys in a family with 3 children, however, tells a slightly different story. The relative frequency histogram estimates the actual proportions considering the natural gender that, compared to the first graph, the chances are greater that a family of 3 will have mostly boys (as indicated by the higher bars for 2 and 3 boys).

4 Although there are slightly more all-boy families than would be predicted by pure probability, the histograms are quite 4 IntroductionProbability DistributionsItIti iss ofoftten predicted that the probability that a newborn baby will be aboy (or girl, as the case may be) is 50%. Due to the natural gender ratio, however, the actual probability that a newborn will be a boy is not exactly 50%. In the United States, for example, the chances are slightly higher that %,p,gyg403530252015105012 Number of boysNumber of Boys in Family(Predicted)30 Probability (%)403530252015105012 Number of boysNumber of Boys in Family(Actual)30 Probability (%)132 CHAPTER 4 RANDOM VARIABLES AND 1329/17/11 2:24 AM9/17/11 2:24 AMCopyright 2011, K12 Inc. All rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the express prior written consent of K12 2011, K12 Inc.

5 All rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the express prior written consent of K12 4 IntroductionApplying ItIn this CHAPTER , you will learn how to create and interpret probability and binomial DISTRIBUTIONS using both tables and histograms. You will work with real data to learn about expected value and continuous RANDOM VARIABLES . You will also learn about normal DISTRIBUTIONS and how the standard normal curve is used to interpret data and make predictions. The Normal DistributionWhen a data set has a normal distribution , a histogram that represents the data set is symmetric and has a bell symmetry of this curve demonstrates that more data are near the mean than in the outlying normal distribution has properties that are very useful in statistics.

6 When data are normally distributed, about 68% of the data values are within a range of 1 standard deviation above and below the example, suppose the average length of a full-term newborn baby is inches with a standard deviation of inches. Because lengths of babies are normally distributed, we can assume that roughly 68% of newborn babies are between and inches in length. These values are obtained by subtracting and adding from Other properties of the normal distribution will be explored later in the ad data set has a normal distribution , a histogram that represents theWhWhdata set is symmetric and has a bell symmetry of this curve demonstrates that more data are near the meanhthi thtl iithaan in the outlying regionsaMeanThe Normal of 4 INTRODUCTION 1339/17/11 2:24 AM9/17/11 2:24 AMCopyright 2011, K12 Inc.

7 All rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the express prior written consent of K12 2011, K12 Inc. All rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the express prior written consent of K12 of Children per Family101286420123456 ChildrenFamiliesPreparing for the ChapterReview the following skills to prepare for the concepts in CHAPTER 4. Find probabilities of events. Interpret histograms. Determine areas of figures in the coordinate plane. Solve linear equations in the form c = x a _____ b . Simplify the expression 1 x for different values of SetA stack of 22 cards are numbered 1 to 22. If a card is randomly selected, find the probability of the event. 1. The number 4 is selected.

8 2. An even number is selected. 3. An odd number less than 8 is selected. 4. A prime number or an even number is selected. 5. A number less than 11 is selected. 6. A number less than 17 or greater than 20 is histogram shows the number of children in 32 selected families. 7. How many families have 5 children? 8. How many families have 2 or more children? 9. How many families have from 1 to 3 children? 10. What percent of the families have from 2 to 4 children?134 CHAPTER 4 RANDOM VARIABLES AND 1349/24/11 1:54 AM9/24/11 1:54 AMCopyright 2011, K12 Inc. All rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the express prior written consent of K12 2011, K12 Inc. All rights reserved.

9 This material may not be reproduced in whole or in part, including illustrations, without the express prior written consent of K12 4 Introductionx110234567234567yx1102345234 56789yx1102345623456789yx110234562345678 91011yDetermine the area of the shaded the value of the value of the expression 1 x for the given value of x. 11. 12. 13. 14. 15. = x 30 _____ 3 16. = x _____ 17. = x 120 _____ 18. = x 450 _____ 12 19. x = 3 __ 8 21. x = 2 __ 9 23. x = x = 22. x = 24. x = 4 INTRODUCTION 1359/17/11 2:29 AM9/17/11 2:29 AMCopyright 2011, K12 Inc. All rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the express prior written consent of K12 2011, K12 Inc. All rights reserved.

10 This material may not be reproduced in whole or in part, including illustrations, without the express prior written consent of K12 910 1920 2930 3940 49f8173354714X0 910 1920 2930 3940 letter X is used to represent a discrete RANDOM variable . For example, when a coin is tossed, the variable X would represent the two outcomes: heads and tails. These outcomes are both discrete and ABOUT ITThe sum of the probabilities in a probability distribution table always equals Probability DistributionsProbabilities of all outcomes of a discrete RANDOM variable can be summarized in a Probability distribution Tables A study of a new treatment for high cholesterol undergoes clinical testing with the results summarized in this frequency this table, X stands for the number of points that the patient s cholesterol level decreased, and f stands for the number of patients.