Confidence intervals

One Sample Interval for the Mean (Z-Interval)

Confidence interval for a population mean μ: if σ known.
Assumes population is Normal or sample size n>30.

sample size n:
sample mean x̄:
population standard deviation σ:

Standard error of the mean (SEM)= σ/√n:

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


There is a CL% chance that [x̄-E,x̄+E] contains the true (population) mean μ. 

Try: 
100 100 10  vs. 1000 100 10   effect of sample size n
100 0 1  vs 1000 0 1 
 
100 0 1  vs 100 0 2   effect of population standard deviation σ

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

σ:
E:
C.L.=1-α
90% Zc=1.645 95% Zc=1.96 98% Zc=2.326 99% Zc=2.576

n= (Zcσ/E)2:round up

Standard error of the mean (SEM) for various s and n
s n
10 30 100 300 1000
1 .3162 .1826 .1 .0577 .0316
10 3.162 1.826 1 .5774 .3162
100 31.62 18.26 10 5.774 3.162