Confidence intervals

One Proportion Interval (1-PropZInt)

sample size n:

#Yeses        OR          Proportion
#Yeses, x:      OR      proportion p̂=x/n:       

q̂:        s:      np: nq: Both should be ≥5

SE(p̂) Standard error of the proportion =√(p̂q̂ / n):

Margin of error:    Maximum likely amount of error (amount by which p̂ differs from p).

Critical values: Zc= 1.645 for 90%, 1.96 for 95%, 2.326 for 98%, 2.576 for 99%

Confidence level α α/2 1-α/2 Critical value= InvNorm(1- α/2)
90% .10 .05 .95 1.645
95% .05 .025 .975 1.960
99% .01 .005 .995 2.575
For any CL, put α/2 as the α in CDF


There is a CL% chance that   p̂±E   [p̂-E,p̂+E]   contains the true proportion p.
MSL: CL% confident that the limits ... actually do contain the true value of the population proportion p.

NB. Quality depends on sample size n; population size N irrelevant. 

Try:
n=100 p=.5  and  n=1000 p=.5   to see effect of sample size.

Minimum sample size n for given p̂, E, and C.L.=1-α

p̂: If no estimate known, use .5
E:
C.L.=1-α
90% Zc=1.645 95% Zc=1.96 98% Zc=2.326 99% Zc=2.576

n=p̂q̂(Zc/E)2:round up

Population size N is irrelevant.

Standard error of the proportion for various p̂ and n
p̂ 1-p̂ √(p̂(1-p̂)) SE(p̂)
10 30 100 300 1000
.01 .99 .1 .032 .018 .01 .0058 .0032
.05 .95 .218 .0689 .0398 .0218 .0126 .0069
.1 .9 .3 .0949 .0548 .03 .0173 .0095
.25 .75 .433 .1369 .0791 .0433 .0250 .0137
.4 .6 .490 .155 .089 .049 .0283 .0155
.5 .5 .5 .1581 .0913 .05 .0289 .0158

Apples:
142 142 143 146 148 148 148 150 150 151 151 152 155 155 158