n!       x       n-x
   -------------  p   (1-p)
     x! ( n-x)!
                                                For this Example:
n = # of trials [# coin tosses)                   n= 10
x = # of times came up heads                      x=9
p = probability given in the null hypothesis      p= .5
1-p =1- probability given                         1-p = 1-.5 =.5
n! = n factorial                                  n!=10!
whatever n is, you multiply by next integer       so 10! 
lower than that, until you get to 1                = l0x9x8x7x6x5x4x3x2x1
x! = x factorial                                  9! = 9x8x7x6x5x4x3x2x1
(n-x)! = (# trials - # times head)!               (n-x)! = (10-9)! = (1)! =1

If you ever have :
0! = 1		   1! =1

Now  plug into formula 7.1 for test statistic:

10!           9    10-9      (10x9x8x7x6x5x4x3x2x1)       9   1
----------- (.5) (.5)       = ------------------------- (.5) (.5)
9!(10-9)!                       (9x8x7x6x5x4x3x2x1)(1)

you don' t have to multiply each #. It's easier if you can bracket 
it off and then cancel like terms (in bold face) .


10x(9!)       9      1            9    1
---------- (.5)  (.5)	  =  10(.5) (.5)
9! (1)

  if the base number is the same, you can add up the exponents!
                  10
           =10 (.5)  =10(.00098) = .0098

* If h1 has "not equal to" we must multiply 
the results by 2,  then we can use this number in the decision rule
* If h1  has ">" or "<" just use the result 
"as-is"  for the decision rule.
·
Notice in Example:
h1: PH   
not equal to .5 -> so multiply   .098 by 2
                 = 2 x .0098
                 = .0196

* Decision Rule e: Reject null hypothesis (0) if test statistic < .05 
   otherwise   do not reject ho

REJECT ho,   since test statistic = .0196 is less than .05 , 
i.e., .0196 < .05 
Reject ho because we have evidence that the null hypothesis 
is not true , we have evidence that coin is not fair.

Why? .0196 = .02  says that there is a 2% chance that a 
fair coin would give you 9 heads  in 10 tosses.    so  it's unlikely
the coin will be fair
   PH  >.5
   PH  <.5  is a  one tailed test - if each of these are by itself
   PH  not equal to .5  is a  two tailed test (two directions)

    Ex. T - maze

            Food                              Sex
                      Left          Right
                            |        |
                               rat            
Binomial - Two admissible outcomes

Let the rat run   n =12 trials
                         x = # of right turns = 8
 PR (Probability of right turn)= .5  
 ho: there is no preference for food or sex
 PR not equal to .5
h1  there is a preference for food or sex.

 Common mistakes on the Quiz
#2) Zy = rZx   This is the Standard Score equation  
    don' t confuse it with

            SDy
    y' = r ----- x      This is the Deviation score equation
            SDx

    * Just because the Standard Deviation is given, 
       it doesn' t mean you have to use it.
    * Don' t get fooled by excess information

I.  Binomial Equation
                        n!       x     n-x
                   -----------  p  (1-p)
                   x! ( n-x)!

This equation gives you the probability of obtaining  
exactly  x " event" in n trials.

Ex.   n= 10 coin tosses
        x = 8 heads
PH  ( probability of heads) = .5
( will always be .5 unless otherwise specified)
    - expect heads to come up 1/2 of the time.
  So you want to determine the probability of obtaining exactly 
8 heads in ten coin tosses the notation to use is:
                 P( x = 8)=

   10!         8    10-8  (10x9x8x7x6x5x4x3x2x1)     8    2     (10)(9)      10
---------- (.5) (.5) 	= ----------------------- (.5) (.5) =   --------  (.5)
8!(10-8)!                (8x7x6x5x4x3x2x1)(2x1)	                  2

			= 45 ( .0009765)	  = .0439
What this means is that you can expect to obtain 8 heads in 
10 tosses 4.39% of the time.

Practice Quiz

#l) In a binomial experiment, a coin is tossed 26 times. 
The coin is biased in favor of tails in that 62% of the 
time it will come up tails. If x is the number of tails,
     Find P( x=12).
     n= 26
     Pr= .62 ( Probability of tails)
     x = 12 tails
     What is the probability of getting exactly 12 tails?  
     P( x= 12)

  26!             12      26-12	
-----------   (.62) (1-.62)   = .3125
12!(26-12)!		    	    

#2)  In research disciplines outside the behavioral 
sciences, the maximum tolerable probability of a type
 I error in a hypothesis test is
    a) .05   b) .01   c) either .05 or .01    d) both .05 and .01	 
    e)  not necessarily .05 or .0l
Answer: E,  not necessarily .05 because .05 is only used in 
                     Behavioral Sciences                          ·

#3)   In determining whether a coin is fair or unfair, an 
      experimenter tosses it twenty  times. The results of 
      this experiment was 11 tails.  The experimenter 
      concludes that the coin is fair.  The experimenter 
      could be making
        a) a type I error     b) a type II error       c) both a) and b)      d) a) or b)
        e) none of the above.
   Answer: b a type II error  because n= 20,   x=11,      " it's fair!"

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