Inferential StatisticsSamples - so far we have been
concerned about describing and summarizing samples or subsets of a population
Inferential stats allows us to “go beyond” our sample and make educated guesses about the population
Inferential Statistics
But, we need some help which comes from Probability theory
Inferential = Descriptive + Probability
Statistics Statistics Theory
What is probability theory and what is its role?
Probability theory, or better “probability theories” are found in mathematics and are interested in questions about unpredictable events
Although there is no universal agreement about what probability is, probability helps us with the possible outcomes of random sampling from populations
Examples “The Chance of Rain” The odds of rolling a five at the craps table (or
winning at black jack) The chance that a radioactive mass will emit a
particle The probability that a coin will come up heads
upon flipping it* The probability of getting exactly 2 heads out of 3
flips of a coin* The probability of getting 2 or more heads out of
3 flips*
Set Theory (a brief digression)Experiment - an act which leads to an
unpredictable, but measurable outcome
Set - a collection of outcomesEvent - one possible outcome; a value
of a variable being measuredSimple probability – the likelihood that
an event occurs in a single random observation
Probability TheoryMost of us understand probability in
terms of a relative frequency measure (remember this, f/n), the frequency of occurrences (f) divided by the total number of trials or observations (n)
However, probability theories are about the properties of probability not whether they are true or not (the determination of a probability can come from a variety of sources)
Relative Frequency (a reminder)
What if we counted all of the marbles in the jar and constructed a frequency distribution?
We find 50 black marbles, 25 red marbles, and 5 white marbles
Relative frequency (proportion) seems like probability
Color f rf
Black 50.50
Red 25.25
White 25 .25Total 100
1.00
Relative Frequency and Probability Distributions A graphical
representation of a relative frequency distribution is also similar to a “Probability Distribution”
00.10.20.30.40.50.60.70.80.9
1
Rel
ativ
e F
req
uen
cy/
Pro
bab
ilit
y
Black
White
Red
What does this mean?
What will happen if we choose a single marble out of the jar?
If we chose 100 marbles from the jar, tallied the color, and replaced them, will we get 50, 25, and 25? If so, what if we selected only 99?
If .5, .25, and .25 are the “real” probabilities, then “in the long run” will should get relative proportions that are close to .5, .25, and .25
Bernoulli’s TheoremThe notion of “in the long run” is
attributed to Bernoulli It is also known as the “law of large
numbers”as the number of times an experiment
is performed approaches infinity (becomes large), the “true” probability of any outcome equals the relative proportion
Axiom’s of Probability
1.The probability of any event A, denoted p(A), is 0 < p(A) < 1
2.The probability of S, or of an event in sample space S is 1
3. If there is a sequence of mutually exclusive events (B1, B2, B3, etc.) and C represents the event “at least one of the Bi’s occurs, then the probability of C is the sum of the probabilities of the Bis (p(C) = Σ p(Bi)
3. p(C) = Σp(Bi)
B1
B3
B4 B2
B5
S
C
If the events B1, B2, B3, etc. are mutually exclusive, the probability of one of the Bs occurring is C, the sum of the Bs
Mutually Exclusive Events If A and B are mutually exclusive,
meaning that an event of type A precludes event B from occurring, by the 3rd axiom of probability
Joint Events If the events are independent, (not
mutually exclusive), meaning that the occurrence of one does not affect the occurrence of the other, the intersection
Joint Events - ExampleWhat is the probability of selecting a
black marble and white marble in two successive selections?
Since each selection is independent, then
p(Black, White) = .5 • .25 = .125
Generalization from Joint Events
If A, B, C, and D are independent events, then:
What is the probability of selecting a white marble, then red, then white, then black?
What if the events are not independent?Conditional probability - the occurrence
of one event is influenced by another event
“Conditional Probability” refers to the probability of one event under the condition that the other event is known to have occurred
p(A | B) - read “the probability of A given that B has occurred”
Probability Theory and Hypothesis TestingA man comes up to you on the street
and says that he has a “special” quarter that, when flipped, comes up heads more often than tails
He says you can buy it from him for $1You say that you want to test the coin
before you buy itHe says “OK”, but you can only flip it 5
times
Probability Theory ExampleHow many heads would convince you
that it was a “special” coin?3?, 4?, 5?How “sure” do you want to be that it is
a “special” coin? What is the chance that he is fooling you and selling a “regular” quarter?
2 HypothesesThe coin is not biased, it’s a normal
quarter that you can get at any bank– The likelihood of getting a heads on a
single flip is 1/2, or .5The coin is a special
– The probability of getting a heads on a single flip is greater than .5
Hypothesis TestingLet’s assume that it is a regular, old
quarterp = .5 (the probability of getting a heads on a SINGLE toss is .5)
We flip the coin and get 4 heads. What is the probability of this result, assuming the coin is fair?
Note that this is a problem involving conditional probability : p(4/5 heads|coin is fair)
How do I solve this problem?Any Ideas?You might think that, using the rule of
Joint events, that:
NO!
Why not?You have just calculated the probability
of getting exactly H, H, H, H on four flips of our coin.
What is the probability of getting H, T, H, T on 4 flips?
Exactly the same as H, H, H, H…any single combination of 4 H and T are equally likely in this scenario.
Here they are:
HHTT
HTHT
TTTH HTTH HHHT
TTHT TTHH HHTH
THTT THTH HTHH
TTTT HTTT THHT THHH HHHH
0 Heads 1 heads 2 heads 3 heads 4 heads
All Possibilities: 4 flips of a coin
f = 1 4 6 4 1
YES!Relative frequency and Probability are
related by Bernoulli’s theorem If I did this test again, would I get the
same result? (probably not) If I did it over and over again, what
results would we expect given a non-biased coin?
How many combinations?
What if I figured out the total number of
possible outcomes of this experiment, and I figured out the total number of
outcomes that had 4/5 heads?
Prob of 4/5 = Freq of 4/5
Total N of outcomesHow many outcomes?
How many 4 out of 5?5 flips(exactly) 4 heads1 possibility – H, H, H, H, TAnother – H, H, H, T, HMore – H, H, T, H, HAnd – H, T, H, H, HLastly – T, H, H, H, H
HHTTT TTHHH
HTTHT THHTH
HTHTT THTHH
THHTT HTTHH
THTHT HTHTH
HTTTT TTHHT HHTTH THHHH
THTTT HTTTH THHHT HTHHH
TTHTT THTTH HTHHT HHTHH
TTTHT TTHTH HHTHT HHHTH
TTTTT TTTTH TTTHH HHHTT HHHHT HHHHH
5 Flips: All possibilities
0 heads 1 head 2 heads 3 heads 4 heads 5 headsp
= .03125 .15625 .3125 .3125 .15625 .03125
At least 4 heads out of 5Given a Fair Coin:Getting at least 4 heads out of 5 flips is
p(4) + p(5)
.15625+.03125 = .1875
There is a 18.75% chance that, upon flipping a FAIR coin 5 times, you will get at least 4 heads.
B1
B3
B4 B2
B5
C