Null Hypotheses, Rejection Regions. and Formulas,

Null hypotheses:
One proportion-- H0 :  p = p0 .  
Two proportions-- H0 :  p1  = p2
Three or more proportions: H0 :  p1  = p2 = p3  = . pk  

One mean: H0 : mu = mu 0.  
Two means: H0 : mu1  = mu2
Three or more means: H0 : mu1  = mu2 =  .  = muk

Rejection Regions (given for the three possible alternatives in the order
a. ' < ' ;  b. ' > '; and c.  ' ¹ ' )

1. One proportion p:
  a.  phat  < c   (that is, if phat is 'small')
  b.  phat  > c   (that is, if phat is 'large')
  c.  phat  < c1 or phat  > c2  (either much smaller or much larger than p0)
  
 *phat is the sample proportion.

2. Two proportions:
  a. p1hat - p2hat  < c
  b. p1hat - p2hat  > c
  c. p1hat - p2hat  < c1 or p1hat - p2hat > c2   

 *p1hat  and  p2hat are the two sample proportions.

3. One population mean mu :
  a. xbar < c
  b. xbar > c
  c. xbar < c1 or xbar > c2   


4. Two population means mu1   and  mu2 :
a. xbar - ybar < c
b. xbar - ybar > c
c. xbar - ybar <c1 or  xbar - ybar > c2

5.       Three or more proportions: reject null hypothesis if p-value is
less than alpha = level  of significance.

6.     Three or more means: reject null hypothesis if p-value is less than
alpha = level of   significance.



Sample mean:   xbar = (sum  x ) / n ;  IQR = Interquartile Range = Q3 - Q1

5# summary:  min   Q1  median  Q3   max

Sample median: Arrange observations in order.  If n is odd, median is
middle observation.  If n is even, median is the average of the middle two.

Sample standard deviation  s:   First calculate s2  = [sum( x2)  - (sum  x
)2 / n ] / (n - 1).  Then s = sqrt (s2 )

x2:  square the x's and then sum them.   (sum x) 2:  add up the x's and get
the sum. then square the sum.

Population mean m : m  = sum[ x p(x)]        where p(x) is the probability
of x.

Population standard deviation  sigma : 

First calculate sigma squared:

sigma squared = sum [ x2 p(x)]  - (mu- squared)  = E(X2) - mu squared .
Then   sigma   = sqrt(sima squared ).

x2:  x squared