1 working with samples the problem of inference how to select cases from a population probabilities...
TRANSCRIPT
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Working with samples
The problem of inference
How to select cases from a population
Probabilities
Basic concepts of probability
Using probabilities
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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?
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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
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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 casesHap-hazard casesSystematic casesRandom cases
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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
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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
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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 themDid we get them from some reason that
would make them not represent the population?
Do they share characteristics not generally found in the population?
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Systematic cases can run afoul of patterns in the
selection criteriaIf 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
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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!
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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
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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
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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
n
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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?
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Basic concepts of probability
Event or trial - the basic thing or process being countedTossing a coinDealing a card
Outcome of event or trial - the characteristic of the event that is notedhead vs. tailsace vs. 2 vs. 3 vs. . . .
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Events
Simple eventsexample, single toss of coinexample, drawing one card from a deck
Compound eventexample, tossing three coinsexample,drawing 5 cards from a deck
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Outcomes of events
Outcomes are characteristics of events
Event - tossing a coinoutcome: heads or tails
Event - drawing a card from a deckoutcome: ace, 2, 3 …outcome: hearts, diamonds, …outcome: king of spades, ...
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Questions
Are the events independent?Yes, if outcome of one event does not
depend upon the outcome of another event.
Consider two coin tossesConsider sex of two children being bornConsider two cards drawn from same deck
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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 eventcoins: p(heads) given heads in first tosschildren: p(boy) given girl in first borncards: p(ace) given ace in first draw
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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)
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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?
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Using probabilities
Multiplication rulep(a & b) = p(a) * p(b|a)example p(h & h) in two tosses of coinexample p(boy & girl) in birth of two
children if events are independent? P(b|a) = p(b)
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Using probabilities (cont.)
Addition rulep(a or b) = p(a) + p(b) - p(a&b)example p(h or t) in coin tossexample p(girl or boy) in birth of childexample p(girl or blue eyes) in childexample p(ace or king) in card drawexample p(ace or heart) in card draw
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Using probabilities (cont.)
Events must be randomCoin must be fairly tossedDeck of cards must be well shuffled
p(red) from urn with 10 red and 90 blackUrn of different color marbles must be well
shaken (not stirred)
These are samples of size one