Working with samples

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

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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