1. + Chapter 6: Modeling Random Events... Normal & Binomial Models

2. + Models... Help us predict what is likely to happen Remember LSRLs (a model for linear, bi-variate data); not exact (some points above line, some below); but best model we have Two more models discussed in this chapter are for Normal (or ≈ Normal) and for Binomial distributions Both of these describe/model numeric, uni-variate data Again, models are not perfect... But in many cases, they are good, effective, or “good enough”

3. + Two types of numeric data... Discrete random variables/data (we will use binomial distribution with discrete data) Continuous random variables/data (we will often use Normal distribution with continuous data)

4. + How are these the same? How are they different? Don’t look ahead! Discrete Random Variable Examples Continuous Random Variable Examples Stars in the skyHours of sleep you got last night Number of burgers you buy at In- N-Out Gallons of gas you put in your car last time you filled up How many coins you have in your pocket Your current weight Times you have gone bowling this month Burgers you ate at In-N-Out Number of pets your parents haveA friend’s age Number of social media platforms you use regularly A co-worker’s height Olympic athlete’s ranking in his/her sport/event Olympic athlete’s best time in his/her sport/event

5. +

6. + Discrete Random Variables Discrete random variables have a “countable” number of possible positive outcomes and must satisfy two requirements. Note: ‘countable’ is not the same as finite. (1) Every probability is a # between 0 and 1; (2) The sum of the probabilities is 1

7. + World-Wide 2015 High School AP Statistics Score Distribution This a discrete random variable probability distribution? Why? Explain. 12345.238.189.252.189.132

8. + Discrete Random Variables... Discuss other examples of discrete random variables. 1 minute.

9. +

10. + Other Examples of Discrete Random Variables... Number of times people have seen Shawn Mendes in concert Number of gifts we get on our birthday Number of taco’s sold at Taco Bell per day Number of stars in the sky All are whole, countable numbers

11. + Non-examples of Discrete Random Variables... Your height Weight of a candy bar Time it takes to run a mile

12. + Continuous Random Variables... are usually measurements heights, weights, time amount of sugar in a granny smith apple, time to finish the New York marathon, height of Mt. Whitney

13. + How can we distinguish between continuous and discrete? Discuss in your groups for a few minutes.

14. + How can we distinguish between continuous and discrete? Discuss in your groups for a few minutes. Ask yourself ‘How many? How much? Are you sure?’ For example, # of children, pounds of Captain Crunch produced each year, # of skittles, ounces in a bag of skittles

15. + Continuous Random Variables... take on all values in an interval of numbers probability distribution is described by a density curve probability of event is area under the density curve and above the values of X that make up the event total area under (density) curve = 1

16. + Continuous Random Variables... Probability distribution is area under the density curve, within an interval, above x-axis

17. + Continuous Random Variables... For all continuous random variables, there is no difference between and ≥ This is not true for discrete random variables. Why?

18. + Random Variables... Consider a six-sided die... What is the probability... P ( roll less than or equal to a 2) is P ( roll less than a 2) is Different probabilities; discrete random variable Note: possible outcomes are 1, 2, 3, 4, 5, 6; but probabilities of those outcomes are (often) fractions/decimals

19. + Continuous random variables... All continuous random variables assign probabilities to intervals All continuous random variables assign a probability of zero to every individual outcome. Why?

20. + Continuous Random Variables... There is no area under a vertical line (sketch) Consider... 0.7900 to 0.8100 P = 0.02 0.7990 to 0.8010 P = 0.002 0.7999 to 0.8001 P = 0.0002 P (an exact value –vs. an interval--) = 0

21. + Density Curves & Continuous RV’s.. Can use ANY density curve to assign probabilities/model continuous distributions/RV’s; many models Most familiar density curves are the Normal (bell) density curves Based on Empirical Rule, 68-95-99.7, symmetric, uni-modal, chapter 3) Many distributions/events are considered Normal & can be modeled by Normal density curves, such as... cholesterol levels in young boys, heights of 3-year-old females, Tiger Woods’ distance golf ball travels on driving range, basic skills vocabulary test scores for 7 th graders, etc.

22. + Mean & Standard Deviation of Normal Distributions... μ x for continuous random variables lies at the center of a symmetrical (or fairly symmetrical) density curve (Normal or approximately Normal) N ( μ, σ ) Remember... μ and σ are population parameters; and s are sample statistics. Calculating σ and/or σ 2 for continuous random variables…. beyond the scope of this course… will be given this information if needed

23. + Normal distributions/density curves...

24. + Four possibilities that we need to know how to calculate... When calculating probabilities for Normal distributions, there are four possibilities that we might be asked to calculate. Let’s draw some pictures and match up some questions to each drawing Remember for continuous random variables (like Normal distributions) there is no difference between ‘less than’ and ‘less than or equal to;’ likewise no difference between ‘greater than’ or ‘greater than or equal to.’

25. + Female heights... N(64.5, 2.5)

26. + Calculate the probability that a randomly chosen female is: shorter than 63 inches no more than 61 inches taller than 66 inches 68 inches or taller between 63.5 inches and 64 inches between 5 inches and 72 inches

27. + Female heights... N(64.5, 2.5) Calculate the probability that a randomly chosen female is: shorter than 63 inches or taller than 70 inches as short or shorter than 65 inches or as tall or taller than 67 inches

28. + Female heights... N(64.5, 2.5) Sometimes we want to turn this around... Sometimes we are given a probability & we need to find the value that corresponds to that probability. What is the height of a randomly chosen woman if she is in the 20 th percentile? Let’s draw a picture... What is the height of a randomly chosen woman if she is in the 85 th percentile?

29. + Normal model... Very helpful, but one size does not fit all Good first choice if data is continuous, uni-modal, symmetric, 68-95-99.7

30. + Another important, “special” type of distribution... If certain criteria is met, easier to calculate probabilities in specific situations Next types of distributions we will examine are situations where there are only two outcomes Win or lose; make a basket or not; boy or girl...

31. + Discuss situations where there are only two outcomes... Yes or no Open or closed Patient has a disease or doesn’t Something is alive or dead Person has a job or doesn’t A part is defective or not

32. + that is what this section is all about...... a class of distributions that are concerned about events that can only have 2 outcomes The Binomial Distribution Binomial Distributions are a special type of discrete random variables

33. + Binary; Independent; fixed Number; probability of Successes The Binomial setting is: 1. Each observation is either a success or a failure (i.e., it’s binary) 2. All n observations are independent 3. Fixed # (n) of observations 4. Probability of success, p, is the same for each observation “BINS”

34. + Binomial Distribution: practice… I roll a die 3 times and observe each roll to see if it is even or odd. Is: 1. each observation is either a success or a failure? 2. all n observations are independent? 3. fixed # (n) of observations? 4. probability of success, p, is the same for each observation ? BINS

35. + Binomial Distribution If BINS is satisfied, then the distribution can be described as B (n, p) Bbinomial nthe fixed number of observations pprobability of success Note: This is a discrete probability distribution. Remember N ( μ, σ )… is that discrete??

36. + Binomial Distribution Most important: being able to recognize situations and then use appropriate tools for that situation Let’s practice...

37. + Are these binomial distributions? Why or why not? Toss a coin 20 times to see how many tails occur. Asking people if they watch ABC news Rolling a die until a 6 appears

38. + Are these binomial distributions or not? Why or why not? Asking 20 people how old they are Drawing 5 cards from a deck for a poker hand Rolling a die until a 5 appears

39. + Side note: Binomial Distribution... Is the situation ‘independent enough’? An engineer chooses a SRS of 20 switches from a shipment of 10,000 switches. Suppose (unknown to the engineer) 12% of switches in the shipment are bad. Not quite a binomial setting. Why? For practical purposes, this behaves like a binomial setting; ‘close enough’ to independence; as long as sample size is small compared to population. Rule of thumb: sample ≤ 10% of population size

40. + Practice... Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that exactly 2 of them have type O blood? Binomial setting? Check for BINS. p = 0.25n = 5X = 2

41. + Practice... Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that exactly 2 of them have type O blood? Binomial setting? Check for BINS. p = 0.25n = 5X = 2 = 0.2636; context, always!

42. + Practice... Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that exactly 4 of them have type O blood? Binomial setting? Check for BINS. p = 0.25n = 5X = 4

43. + Practice... Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that exactly 4 of them have type O blood? Binomial setting? Check for BINS. p = 0.25n = 5X = 4 = 0.0146; context, always

44. + Practice... Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that exactly 1 of them have type O blood? Binomial setting? Check for BINS. p = 0.25n = 5X = 1

45. + Practice... Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that exactly 1 of them have type O blood? Binomial setting? Check for BINS. p = 0.25n = 5X = 1 = 0.3955; context, always

46. + Practice... Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that at most 2 of them have type O blood? Binomial setting? Check for BINS. p = 0.25n = 5X = 2

47. + Practice... Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that at most 2 of them have type O blood? Binomial setting? Check for BINS. p = 0.25n = 5X = 2 = 0.8965; context, always

48. + Practice... Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that at most 4 of them have type O blood? Binomial setting? Check for BINS. p = 0.25n = 5X = 4

49. + Practice... Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that at most 4 of them have type O blood? Binomial setting? Check for BINS. p = 0.25n = 5X = 4 = 0.9990, context, always

50. + Practice... Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that at most 1 of them have type O blood? Binomial setting? Check for BINS. p = 0.25n = 5X = 1

51. + Practice... Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that at most 1 of them have type O blood? Binomial setting? Check for BINS. p = 0.25n = 5X = 1 = 0.6328, context, always

52. + Practice...... probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that at least 2 (meaning 2, 3, 4, or 5) of them have type O blood? p = 0.25n = 5X = 2, 3, 4, or 5

53. + Practice...... probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that at least 2 (meaning 2, 3, 4, or 5) of them have type O blood? p = 0.25n = 5X = 2, 3, 4, or 5 1 - 0.6328 = 0.3672; context, always

54. + Practice...... probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that at least 3 (meaning 3, 4, or 5) of them have type O blood? p = 0.25n = 5X = 3, 4, or 5

55. + Practice...... probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that at least 3 (meaning 3, 4, or 5) of them have type O blood? p = 0.25n = 5X = 3, 4, or 5 1 - 0.8965 = 0.1035; context, always

56. + Practice...... probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that more than 3 (meaning 4 or or 5) of them have type O blood? p = 0.25n = 5X = 4 or 5

57. + Practice...... probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that more than 3 (meaning 4 or or 5) of them have type O blood? p = 0.25n = 5X = 4 or 5 1 - 0.9844 = 0.0156; context, always

58. + Practice...... probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that … of them have type O blood? p = 0.25n = 5 a) …at most 3 of them… b) …at least 4 of them… c) …more than 1 of them… d) …exceeds 3 of them… e) … below 2 of them… f) … 0 of them…

59. + Extra practice with binomials?? Page 285 if needed

60. + Binomial Distribution: Mean and Standard Deviation If a basketball player makes 75% (“p” of her free throws, what do you think the mean number of baskets made will be in 12 tries?

61. + Binomial Distribution: Mean and Standard Deviation If a basketball player makes 75% (“p” of her free throws, what do you think the mean number of baskets made will be in 12 tries? (0.75) (12) = 9; we expect she should make 9 baskets in 12 tries. Her mean number of baskets made should be 9. We expect her = 9 baskets; E(x) = 9

62. + Binomial Mean & Standard Deviation If a count X is a binomial distribution with number of observations n and probability of success p, then μ = np and σ = Only for use with binomial distributions; remember criteria... BINS

63. + Practice... If a basketball player makes 75% (“p”) of her free throws, we expect her to make 9 baskets in 12 tries. What is the SD of this distribution?

64. + Practice... If a basketball player makes 75% (“p”) of her free throws, we expect her to make 9 baskets in 12 tries. What is the SD of this distribution? SD = = = 1.5

65. + Practice... Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the mean and standard deviation for this distribution?

66. + Practice... Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the mean and standard deviation for this distribution? mean = np = (5)(.25) = 1.25; the family should expect to have 1.25 children with type O blood SD = = = 0.9682

67. + Remember & Caution... Binomial distribution is a special case of a probability distribution for a discrete random variable All binomials distributions are discrete random variable distributions BUT not all discrete random variable distributions are binomial distributions Don’t assume; don’t apply binomial tools to all discrete RV distributions; must meet binomial distribution criteria (BINS)

68. + Next test... Exam #2... Chapters 4, 5, & 6