1. 1 Working with samples The problem of inference How to select cases from a population Probabilities Basic concepts of probability Using probabilities

2. 2 The problem of inference We work with a sample of cases from a population We are interested in the population We would like to make statements about the population, but we only know the sample Can we generalize our finding to the population?

3. 3 We can generalize Under certain conditions If we make certain assumptions If we follow certain procedures If we don’t mind being wrong a certain percentage of the time

4. 4 How to select cases from a population The first condition for generalization is to select our cases from the population in a certain way. What ways are possible? Representative cases Hap-hazard cases Systematic cases Random cases

5. 5 We choose random cases Because we can use probability theory to help us know the unknowable. Representative cases are nice, but how do we know they are representative? Hap-hazard cases are the worst and we will see why. Systematic cases can run afoul of patterns in the selection criteria

6. 6 How do we know if cases are representative? To know if a case is representative of the population, we must already know the population! But, we are trying to find out about the population

7. 7 Hap-hazard cases are the worst We don’t know if they represent the population We don’t know the reasons we came to select them Did we get them from some reason that would make them not represent the population? Do they share characteristics not generally found in the population?

8. 8 Systematic cases can run afoul of patterns in the selection criteria If we have a list of the members of the population and take every 10th case: What if we are sampling workers and a foreman is listed followed by the 9 people under them

9. 9 Random samples are the best We can use probability theory, because random is a probability concept Probability theory is a branch of mathematics, and it can get very hairy But, not in this class Only addition, subtraction, multiplication, and division, as always, are used -- and you can do that!

10. 10 Probabilities Probabilities are hypothetical, but very helpful Probabilities are numbers between 0.0 and 1.0 A probability is a relative frequency in the long run

11. 11 Probabilities (cont.) Relative frequency is like a proportion A proportion is f/n expressed as a decimal number (e.g.,.4) For example, the probability it will rain today is.95 This means that on 95/100 days like this we expect it to rain

12. 12 Probabilities (cont.) But, do we look at 100 days? Should we base this prediction on 1000 days? In the long run refers to the idea that we may let the number of days That is let the number of trials approach infinity, or all imaginably possible

13. 13 Probabilities (cont.) What is the probability of getting a heads on a fair toss of a coin? What is the probability of drawing a red ball from a jar containing 1 red and 3 black balls?

14. 14 Basic concepts of probability Event or trial - the basic thing or process being counted Tossing a coin Dealing a card Outcome of event or trial - the characteristic of the event that is noted head vs. tails ace vs. 2 vs. 3 vs....

15. 15 Events Simple events example, single toss of coin example, drawing one card from a deck Compound event example, tossing three coins example,drawing 5 cards from a deck

16. 16 Outcomes of events Outcomes are characteristics of events Event - tossing a coin outcome: heads or tails Event - drawing a card from a deck outcome: ace, 2, 3 … outcome: hearts, diamonds, … outcome: king of spades,...

17. 17 Questions Are the events independent? Yes, if outcome of one event does not depend upon the outcome of another event. Consider two coin tosses Consider sex of two children being born Consider two cards drawn from same deck

18. 18 Independence Two events are independent if p(x) -- the probability of x -- in the second event does not depend upon the p(x) in the first event coins: p(heads) given heads in first toss children: p(boy) given girl in first born cards: p(ace) given ace in first draw

19. 19 Conditional probabilities Drawing 2 cards (without replacement) p(ace) in second card given ace in first, written as p(a|a) p(ace) in second card given king in first, written as p(a|k) Independence requires p(a) = p(a|a) and p(a) = p(a|k)

20. 20 Questions (cont.) Are the events mutually exclusive? Yes, if the two events cannot occur together Is the birth of a male first child exclusive of the birth of a female first child? Is the birth of a male first child exclusive of the birth of a child with brown hair?

21. 21 Using probabilities Multiplication rule p(a & b) = p(a) * p(b|a) example p(h & h) in two tosses of coin example p(boy & girl) in birth of two children if events are independent? P(b|a) = p(b)

22. 22 Using probabilities (cont.) Addition rule p(a or b) = p(a) + p(b) - p(a&b) example p(h or t) in coin toss example p(girl or boy) in birth of child example p(girl or blue eyes) in child example p(ace or king) in card draw example p(ace or heart) in card draw

23. 23 Using probabilities (cont.) Events must be random Coin must be fairly tossed Deck of cards must be well shuffled p(red) from urn with 10 red and 90 black Urn of different color marbles must be well shaken (not stirred) These are samples of size one