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HYPOTHESIS TESTS ON MEANS AND PROPORTIONS

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					             HYPOTHESIS TESTS ON MEANS AND PROPORTIONS


(1)   population mean - 1 sample - known population variance:
                                     - - 
                                     X
                              Z =   ----------

                                      /n

(2)   population mean - 1 sample - unknown population variance:
                                     - - 
                                     X
                              t =   ----------

                                      s/n

                              d.o.f. = n - 1


(3)   difference in population means - 2 independent samples -
      known population variances:
                                - -
                               (X1-X2) - (1-2)
                       Z =     -----------------
                                 _______________
                                 12/n1 + 22/n2


(4)   difference in population means - 2 independent samples -
      unknown population variances:
                                - -
                               (X1-X2) - (1-2)
                       t =     -----------------
                                 _______________
                                s12/n1 + s22/n2


                       d.o.f. = min(n1, n2) - 1


(5)   difference in population means - 2 independent samples -
      unknown population variances that are believed to be equal:
                                - -
                               (X1-X2) - (1-2)
                       t =     -----------------
                                 ________________
                                sp2(1/n1 + 1/n2)


                               (n1-1)s12 + (n2-1)s22
                      sp2 =    ---------------------
                                    n1 + n2 - 2


                       d.o.f. = n1 + n2 - 2
(6)   difference in population means - 2 dependent samples:
                                   -
                                   D - 
                             t =   ---------

                                   sD/   n

                           d.o.f. = n - 1
           where n is the number of pairs of observations


(7)   population proportion - 1 sample - (at least 100
      observations):
                                    - - p
                                    p
                             Z =    -----
                                    pq/n


(8)   difference in population proportions - 2 independent samples
      - (at least 100 observations in each sample):

                               - -
                              (p1-p2) - (p1-p2)
                       Z =    -----------------
                                --
                                pq(1/n1 + 1/n2)

                             -
                             p = (X1 + X2)/(n1+n2)

                             -       -
                             q = 1 - p
           CONFIDENCE INTERVALS FOR MEANS AND PROPORTIONS


(1)   population mean - known population variance:

                   _        _                  _         _
                   X - Z /n    < X + Z/n


(2)   population mean - unknown population variance:

                   _        _                  _          _
                   X - t s/n    < X + ts/n


                                 d.o.f. = n - 1


(3)   difference in population means - known population
      variances:

 _ _                                                _ _
(X1-X2)- 1 2/n1 + 22/n2 < - 2< (X1-X2) + 12/n1 + 22/n2



(4)   difference in population means - unknown population variances:

 _ _                                             _ _
(X1-X2)-t s12/n1 + s22/n2 < - 2< (X1-X2) +t     s12/n1 + s22/n2



                          d.o.f. = min(n1, n2) - 1


(5)   difference in population means -
      unknown population variances that are believed to be equal:

 _ _                                               _ _
(X1-X2)-tsp 2(1/n1 + 1/n2) < - 2< (X1-X2) +tsp2 (1/n1 + 1/n2)


                                 (n1-1)s12 + (n2-1)s22
                         sp2 =   ---------------------
                                      n1 + n2 - 2


                          d.o.f. = n1 + n2 - 2
(6)   population proportion (at least 100 observations):

                   -      - -                 -       - -
                   p -Z   pq/n    <   p   <   p + Z   pq/n

                            -       -
                            q = 1 - p



(7)   difference in population proportions
      (at least 100 observations in each sample):


 - -       - -       - -                    - -       - -    - -
(p1-p2)- Z p1q1/n1 + p2q2/n2     < p1-p2 < (p1-p2)+ Z
p1q1/n1+p2q2/n2


                            -       -
                            q = 1 - p