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

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

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

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  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!

  5. 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

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

  7. Discrete Random Variables... • Discuss other examples of discrete random variables. 1 minute.

  8. 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

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

  10. 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

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

  12. 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

  13. 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

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

  15. Continuous Random Variables ... • For all continuous random variables, there is no difference between < and ≤... No difference between > and ≥ • This is not true for discrete random variables. Why?

  16. 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

  17. 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?

  18. 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

  19. 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 7th graders, etc.

  20. Mean & Standard Deviation of Normal Distributions... • μx for continuous random variables lies at the center of a symmetrical (or fairly symmetrical) density curve (Normal or approximately Normal) • N (μ, σ) • Remember... μandσ are population parameters; and s are sample statistics. • Calculating σ and/or σ2 for continuous random variables…. beyond the scope of this course… will be given this information if needed

  21. Normal distributions/density curves...

  22. 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.’

  23. Female heights... N(64.5, 2.5)

  24. Female heights... N(64.5, 2.5) • 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

  25. 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

  26. 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 20th percentile? Let’s draw a picture... • What is the height of a randomly chosen woman if she is in the 85th percentile?

  27. 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

  28. 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 ...

  29. 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

  30. 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

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

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

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

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

  35. 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

  36. 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

  37. 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

  38. 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 = 0.25 n = 5 X = 2

  39. 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 = 0.25 n = 5 X = 2 = 0.2636; context, always!

  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 4 of them have type O blood? Binomial setting? Check for BINS. p = 0.25 n = 5 X = 4

  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 4 of them have type O blood? Binomial setting? Check for BINS. p = 0.25 n = 5 X = 4 = 0.0146; 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 1 of them have type O blood? Binomial setting? Check for BINS. p = 0.25 n = 5 X = 1

  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 1 of them have type O blood? Binomial setting? Check for BINS. p = 0.25 n = 5 X = 1 = 0.3955; 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 at most 2 of them have type O blood? Binomial setting? Check for BINS. p = 0.25 n = 5 X = 2

  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 at most 2 of them have type O blood? Binomial setting? Check for BINS. p = 0.25 n = 5 X = 2 = 0.8965; 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 4 of them have type O blood? Binomial setting? Check for BINS. p = 0.25 n = 5 X = 4

  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 4 of them have type O blood? Binomial setting? Check for BINS. p = 0.25 n = 5 X = 4 = 0.9990, 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 1 of them have type O blood? Binomial setting? Check for BINS. p = 0.25 n = 5 X = 1

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