1. Chapter 11 Understanding Randomness

2. Practical Randomness Suppose a cereal company puts pictures of athletes on cards in boxes of cereal in hopes to boost sales. The company announces that 20% of boxes contain a picture of Tiger Woods, 30% a picture of Lance Armstrong, and the rest (50%) will have a picture of Serena Williams. You want all three cards. How many boxes of cereal do you expect to buy to get all three cards?

3. Set Up a Model We will use the digits 0, 1, 2, 3,….9 Assuming they are equally likely to come up and we don’t know which one will be next Tiger Woods is in 20% of the boxes so he should be represented by 20% of the digits we use Lance Armstrong is in 30% of the boxes so he should be represented by 30% of the digits we use Selena Williams is in 50% of the boxes so she should be represented by 50% of the digits we use

4. A Simulation: consists of a sequence of random outcomes that model a situation Vocabulary Component: the most basic part of a simulation Outcomes: each component has a set of possible outcomes Trial: the sequence of events we want to investigate Response Variable: what happened Example Buying a box of cereal The card that is in the box Getting all three cards How many boxes did we need to buy to get all three cards

5. Steps To Making A Simulation 1. Identify the component to be repeated –I–In this case, our component is the selection of a box of cereal. 2. Explain how you will model the outcomes. –T–The digits 0 to 9 are equally likely to occur. Because 20% of the boxes contain Tiger’s picture, we’ll use 2 of the ten digits to represent that outcome. Three of the ten digits can model the 30% of boxes that contains Lance cards, and the remaining 5 digits can represent the 50% of boxes with Serena. 0, 1 = Tiger 2, 3, 4 = Lance 5, 6, 7, 8, 9 = Serena

6. Steps Cont. 3. Explain how you will simulate the trial. – A trial is the sequence of events that we are pretending will take place. In this case we want to pretend to open cereal boxes until we have one of each picture. We do this by looking at each random number and indicating what outcome it represents. We continue until we have encountered all three pictures Example: the sequence 29240 – 2 : Lance – 9 : Serena – 2 : Lance – 4 : Lance – 0 : Tiger Since we got all three pictures that is the end of a trial

7. Steps Cont 4. State clearly what the response variable is – What are we interested in? We want to know how many boxes it takes to get all three pictures. This is the response variable. In this sample trial here, the response variable is 5 boxes. 5. Run several trials – A simulation is cheaper than really buying cereal, and the more trials you perform, the better.

8. Last Steps 6. Analyze the response variable. – We wanted to know how many boxes we might expect to buy to get all three cards. To answer the question, we need to analyze the response variable. Mean: 7.8 boxes for the first 5 trials 7. State your conclusion. – Based on our simulation, we estimate that customers who want the complete set of sports star pictures will by an average of 7.8 boxes of cereal. We only ran 5 trials which is not enough (but for example sake it works) You should run at least 20 trials when working by hand You should run a few hundred when using a computer

9. Chapter 12 Sample Surveys

10. Sampling We use sampling to get an idea about the whole population with out asking the entire population. We take what we know about the sample and stretch that over everyone To do this we have three ideas

11. Idea 1: Examine a Part of the Whole Draw a sample – It is impractical or sometimes impossible to survey the entire population – We examine a smaller group of the population called a sample A small sample (if selected properly) can represent the entire population

12. Sample Surveys Opinion polls  designed to ask questions of a small group of people in hopes of learning something about the entire population If the sample does not represent the population the information can be misleading

13. Bias When selecting a sample you want to make sure that you are not over- or under- emphasizing some characteristics of the population How will you select your sample?? – Phone number list?? homeless people without a land line – Internet Surveys??? people that don’t have internet

14. Bias Sampling methods that, by their nature, tend to over- or under- emphasize some characteristics of the population are bias – Voluntary response samples: people choose themselves – Convenience samples: your sample is made up of people close by It is the most important thing to avoid when sampling – the data and conclusions will be flawed

15. Idea 2: Randomizing Randomizing protects us from the influences of all the features of our population, even ones that we may not have thought about. It does that by making sure that on average the sample looks like the rest of the population

16. Idea 3: It’s the Sample Size How big should the sample be? – The number of individuals in the sample is all that matters – It has very little to nothing to do with the size of the population The fraction of the population that you’ve sampled does not matter. It’s the sample size itself that’s important Surprising?!?! YES!!! – But very important – It balances between how well the survey can measure the population and how much the survey costs For a survey that tries to find the proportion of the population that falls into a category you would need at least a few hundred individuals

17. Census A survey to the entire population What factors make a census difficult? – difficult to complete some people are hard to find – populations rarely stand still people die, move, babies are born, opinions change – more complicated that sampling team effort, population needs to cooperate US Census records too many college students because they are being counted twice (home and school)