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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