# 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

```
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