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 Two more models discussed in this chapter for Normal (or at least ≈ Normal) and for Binomial distributions Both of these describe/model numerical data Again, models are not perfect... But in many cases, they are good or “good enough”

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) Let’s discuss discrete random variables/data first

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 Two types of RV’s we will discuss: discrete and continuous; First we will discuss discrete. Note: countable is not the same as finite. Countable in the sense that we can list successive outcomes together with their associated probabilities. So, for example, the # of stars in the sky is discrete, even though, arguably, # of stars is infinite. Discuss the above example. Values can only be 0,1,2,3,4. No fractions/decimals; but prob’s can be fractional; in fact are the majority of the time as they must sum to 1.

World-Wide 2015 AP Statistics Score Distribution Is this a discrete random variable probability distribution? Why or why not? 1 2 3 4 5 .238 .189 .252 .132

Discrete Random Variables... Discuss other examples of discrete random variables. 1 minute. Think-Pair-Share

Other Examples of Discrete Random Variables... Number of times people have seen Fall Out Boy in concert Number of gifts we get on our birthday Number of burgers sold at In-N-Out Burger per day Number of stars in the sky All are whole, countable numbers; all vary; usually represented by a table or probability histogram

Non-examples of Discrete Random Variables... Your height Weight of a candy bar Time it takes to run a mile These are continuous RV’s which will be discussed in a few; Also can discuss shoe sizes vs. foot length; shoes come in sizes (for example) 5, 5-1/2, 6, 6-1/2, etc. whereas foot length has an infinite number of possibilities. also time to run vs. # hurdles cleared; also age vs. # of birthdays you have had.

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 Are we ‘sure’ that the given runner’s time was 4 hours and 36 minutes? We know the exact time only as accurately as the rounding we choose to use. Discuss height in this manner (for Mt. Whitney; by the way, Mt. Whitney is said to be 14, 494 feet tall), discuss grams of sugar in a granny smith apple.

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 TPS; Pounds of CC vs. # of cereal bits of CC; remind students about discrete does not necessarily mean finite.

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 Bring back shoe size vs. length of foot; time vs. # of hurdles cleared; age vs. # birthdays

Continuous Random Variables... Probability distribution is area under the density curve, within an interval, above x-axis These happen to be uniform density curves. Another example of a uniform density curve is in your text page 248. There are many density curves of all shapes. Calculating an actual density curve is beyond the scope of this course. You will be given a density curve or told what the distribution is.

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?

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

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? Not the case for discrete. It is possible to find the prob of an individual outcome; think of rolling a 3 on a 6-sided die.

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 Could also talk about a limit 1/infinity = 0

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. Z-score work we did previously and histogram work.. Coming back now...

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... μand σ 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 Most often Mu will be given to you as well. Go over N (Mu, sigma) notation.

Normal distributions/density curves... Exact shape depends on SD

Female heights... N(64.5, 2.5) True population parameter for young adult female heights has mu = 64.5 inches and sigma = 2.5 inches. Practice; prob a female being ... Shorter than, taller than, between, etc. START with easy #s 1 SD, 2 SD’s etc. NEXT harder, like between 60” and 63”, etc. all using Minitab. MINITAB: prob, dist plot, display prob, a specified x value, choose picture. Ask about a really short adult female, then use our students for calculations as well. THEN do some inverse norm using same place in minitab.

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 Discuss both; left Normal is good model; not exact, but very good; and talk about width of columns going to 0 (spiral); right graph: right skewed, not Normal; Normal not a good model to use.

Another important, “special” types 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 ...

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 TPS

that is what this section is all about... ... a class of distributions that are concerned about events that can only have 2 outcomes Binomial Distribution

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” Die; even or odd; 2 rolls; prob same; independent rolls; check.

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 Die; even or odd; 2 rolls; prob same; independent rolls; check.

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 this discrete??

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

Are these binomial distributions? Why or why not? Toss a coin 20 times to see how many tails occur. Asking 200 people if they watch ABC news Rolling a die until a 6 appears Yes, yes, no

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 No, no, no

How could we change the situations to make these/force these to be binomial distributions? Asking 20 people how old they are Drawing 5 cards from a deck for a poker hand Rolling a die until a 5 appears

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

Binomial probabilities: Probability Density Function... Use for exact value (i.e., probability that a family has exactly 2 boys); ‘p’ point, particular place Remember, these are discrete random variables Can calculate probabilities of exact values (unlike continuous random variables) Minitab: go to Probability, Probability Density Function, x =, binomial, trials, event probability P, pointing, particular prob, prob density function, value is what x =, choose binomial, trials, event prob.

Binomial probabilities: Cumulative Distribution Function... Use for cumulative values (i.e., probability that a family has at most 2 boys; at least 3 girls; no more than 4 boys, etc.) Cumulative to left; at that value or less If want area to right, need to do “1 minus ....” Pay attention to < vs. ≤; and > vs. ≥ Minitab go to Probability, Cumulative Distribution Function, value (what x = ), binomial, trials, event probability

Practice... *** start here *** 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! Not sure if it will be child #1 & #2 or #1 & #5 or #4 & #3....etc. Just exactly 2 but not sure which 2

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

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

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 Minitab, prob, cum dist function, ...

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

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 = 0.6328, context, always

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 = 0.25 n = 5 X = 2, 3, 4, or 5 1 - 0.6328 = 0.3672; context, always

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 = 0.25 n = 5 X = 3, 4, or 5 1 - 0.8965 = 0.1035; context, always

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 = 0.25 n = 5 X = 4 or 5 1 - 0.9844 = 0.0156; context, always

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 = 0.25 n = 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…

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

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 Emphasize we don’t want to use these formulas with any RVs… they must be binomial RVs

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

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 Emphasize that binomials distributions are count data; can’t be fractions/decimals; but MEANS of discrete RV (binomials) CAN be fractions/decimals.

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)

Next test... Chapters 4, 5, & 6