Two sample, independent, pooled, Proportions hypothesis test (2-Prop Z-Test) and confidence intervals (2-Prop Z-Interval)

H0: p1 = p2
Choose one:
HA: p1<p2 "HA < H0" Left-tailed
HA: p1>p2 "HA > H0" Right-tailed
HA: p1≠p2 "HA ≠ H0" Two-tailed
n1: n2:
#Yeses                                    OR          Proportion
#Yeses, x1: x2:      OR      proportion p̂1=x1/n1:    p̂2=x2/n2:

q̂1=(1-p̂1): q̂2=(1-p̂2):

n1p̂1: n1q̂1:    n2p̂2: n2q̂2: All should be ≥5

p̄ (pooled sample proportion=(x1+x2)/(n1+n2): A kind of average proportion. If same Pop, is the common proportion.   
q̄=1-p̄:

z:      Two-tailed Critical values: Zc= ±1.645 for 90%, ±1.96 for 95%, ±2.326 for 98%, ±2.576 for 99%

p-value (CDF(z)):

p̂1-p̂2:

Confidence intervals for the difference between population proportions: p1-p2

There is a CL% chance that [(p̂1-p̂2)-E, (p̂1-p̂2)+E] contains the true difference of the populations' proportions p1-p2.
** If CI does not contain 0, reject H0, i.e. there is a significant difference between the two populations..
** If CI contains 0, fail to reject H0. 

e-cigs: successes 79 / 438
patch: successes  44 / 446

b-ball: home wins 127 / 198 games
f-ball: home wins 57 / 99 games
H0:p1=p2   Ha:p1!=p2     p_value=.27

splint: successes 60 / 83 treatments
surgery:    "     67 / 73    "

tennis call challenges
M: 1757 / 6036
F:  887 / 3327

Salk polio vaccine
vaxxed: 33 / 200000
unvax: 115 / 200000

Dog socks
123 / 175  w/malaria  
131 / 145  w/o malaria