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 dont 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 dont know if they represent the population
• We dont 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(aa)
• p(ace) in second card given king in first,
  written as p(ak)
• Independence requires p(a) p(aa) and p(a)
  p(ak)

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(ba)
• example p(h h) in two tosses of coin
• example p(boy girl) in birth of two children
• if events are independent? P(ba) p(b)

22
Using probabilities (cont.)

• Addition rule
• p(a or b) p(a) p(b) - p(ab)
• 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