Statistical Arguments

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Statistical Arguments
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Inductive Generalization from the particular to
the general Sampling Arguments.     
Statistical Generalisations some specific
proportion of the members of the target class
possess a certain property.             10 of
people in this sample of the general population
indicated they would vote for Bob.
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------------------------------10 of the general
population would vote for Bob.
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Evaluating Inductive Generalisation   1. Are
the premises true?   hearsay or popular
opinion and not fact? people lie to pollsters.
  2. Is the sample large enough?          
75 of observed coin tosses come up heads
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----- 75 of all coin tosses will come up heads
  four coin tosses? 400 tosses?  
a. Fallacy of Hasty Generalisation  
b. Fallacy of anecdotal evidence.
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Evaluating Inductive Generalisation   3. Is
the sample biased in some other way?   Fallacy
of Biased Sampling.   a. Insufficient variation
in the sample           1. Using a phone book
to get a large sample of the population.  
2. Using birth-order as a behavioural predictor
  3. Basing inferences concerning the general
behaviour of native pigeons, say, on their
behaviour when you've observed them.
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Evaluating Inductive Generalisation  
b. Eliciting a particular characteristic from
the sample by Slanted Questions   1.
Washington Post-ABC News Poll April 2005   36.
Would you support or oppose changing Senate rules
to make it easier for the Republicans to confirm
Bushs judicial nominees?            
Support                Oppose                 No
Opin.           26                        
66                         8   2       
stating they would Sample Conservatives Moderates Liberals
Change procedures to make sure the full Senate gets to vote, up-or-down, on every judge the President nominates. 64 78 59 42
 
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Evaluating Inductive Generalisation   4. Is the
inference justified?   i. The sample size  
ii. The level of confidence   iii. The
margin of error
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Inductive Particularisations - from statistical
generalities to facts or claims about particulars
   Arguments from Statistical Premisses, or
Statistical Applications.             All
ravens we have seen have been black          
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         The next raven we will see will be
black             72 of all Australians are
content with their lives           Robert is an
Australian           ----------------------------
--------------------------------          
Robert is content with his life            
Most Australians are happy           Bob is an
Australian           ----------------------------
------         Bob is happy
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Evaluating Inductive Particularisations  
1.       Are the premises true?   2.       How
Strong is the Conclusion?   3.       Is the
Reference Class the Appropriate One?           
98 of people who have their gall bladder removed
recover easily           Martha is going to have
her gall bladder removed          
--------------------------------------------------
---------------------------           Martha
will recover easily            98 of 90 year
old people who have surgery do not recover easily
          Martha is a 90 year old person about
to have surgery           -----------------------
--------------------------------------------------
----           Martha will not recover easily  
The reference class must meet the requirement of
total available relevant evidence.
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Note on Terms   Consider the following five
quantities 180, 40, 25, 15, 15.   i. The mean
is the arithmetic average. To find it add the
numbers together and divide by 5. Result 55.
  ii. The median is the number in the middle of
the range (1/2 of the numbers are bigger than it
and ½ are smaller). Result 25.   iii. The
mode is the number that is most common. Result
15.
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Numerical illiteracy innumeracy leads to
acceptance of claims based on bad maths  
Example AIDS Testing    (HYP)        Assume
that there is a test for AIDS that is 98
accurate. I.e. if x has AIDS then x will test
positive 98 of the time and if x doesn't have
AIDS x will test negative 98 of the time.  
Assume also that 0.5 of the population of
Australia has the virus i.e., one person in
every two hundred, on average.   Assume 10,000
tests are carried out.   Number of people in
(average) sample having AIDS 0.5 of 10,000
50. Of these, 98 will test positive (by
HYP)                                           
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----------------------------------- Number of
people having AIDS testing positive 98 of 50
49.   Number of people in (average) sample
not having AIDS 99.5 of 10,000 9,950. Of
these, 2 will test positive (by HYP since 98
will test negative)                  -------------
--------------------------------------------------
--------------------- Number of people not
having AIDS testing positive 2 of 9,950
199.   248 people in every (average) population
sample of 10,000 will test positive yet only 49
of these have the disease. Less than 1/5th of
all those testing positive have good reason to
worry!   if you have AIDS then there's a 98
chance of you testing positive TRUE   if you
test positive then there's a 98 chance that you
have AIDS FALSE   if you test positive then
there's less than a 20 chance of you having the
disease TRUE
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numerical illiteracy   Example Psychic
Phenomena  
Psychic Phenomena
Extra-sensory perception
Psychokinesis
Telepathy
Clairvoyance
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numerical illiteracy   a.                
Predictive Dreams and Precognition  
Chance/night of predictive dream 1/10,000.
Chance/night of not predictive dream 1 -
1/10,000 9,999/10,000   Chance of two
successive nights of non-predictive dreams is
          9,999/10,000 x 9,999/10,000   Chance
of only non-predictive dreams all year is
          (9,999/10,000)365 0.964.   So
3.6 do have a predictive dream by chance on some
night of the year.
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For 18 million people we should expect about
650,000 predictive dreams each year. The
following is thus bad numerical reasoning  
(1) The probability of a predictive dream
occurring by chance is so low that the number
of actual instances cannot plausibly be put
down to coincidence. So, (2) Precognition is a
more plausible explanation. Yet, (3) Either
predictive dreams happen by chance or
precognition occurs. So, (4) The most
plausible explanation of predictive dreams is
that precognition occurs.
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Numerical illiteracy   b. Telepathy on Demand
  (1) The probability of cold reading
occurring by chance is so low that the number
of actual instances cannot plausibly be put
down to coincidence. So, (2) Telepathy is a more
plausible explanation. Yet, (3) Either cold
reading happens by chance or telepathy occurs
So, (4) The most plausible explanation of cold
reading is that telepathy occurs.
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Numerical illiteracy   c. General Schematic
Argument   (1) The probability of an X
occurring by chance is so low that the number
of instances cannot plausibly be put down to
coincidence. So, (2) Y is a more plausible
explanation. Yet, (3) Either X happens by chance
or Y. So, (4) The most plausible explanation of
X is that Y.
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Numerical illiteracy   Example Gamblers
Fallacy   The fallacy of thinking that since,
say, a fair coin has come up heads eight times in
a row it is more likely to come up tails next
throw.