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Last modified: January 26, 2018

Confidence Intervals
--------------------

An interval estimate is defined by two numbers, between which 
a population parameter is said to lie. For example, a < μ < b 
is an interval estimate for the population mean, μ. 

Confidence intervals are used to describe the amount of 
uncertainty associated with a sample estimate of a population 
parameter.

The confidence level describes how well the sample statistic 
estimates the underlying population value.  Suppose we used the 
same sampling method to select different samples and to compute 
a different interval estimate for each sample.  Some interval 
estimates would include the true population parameter and some 
would not. A 90% confidence level means that we would expect 
90% of the interval estimates to include the population 
parameter.  

The other way to say this is that if we drew repeated samples 
90% of them would be sure to contain the population mean.  In 
other words, although we don't know the population mean we're 
90% sure it somewhere in the interval. 

The CI does not mean there is a 90% chance that the population 
parameter falls between a and b. This is incorrect.  The 
population parameter is a constant not a random variable.  The 
probability that it falls within a given range is either 0 or 1.

In the same way that statistical tests can be one or two-sided, 
confidence intervals can be one or two-sided. A two-sided 
confidence interval brackets the population parameter from 
above and below.  A one-sided confidence interval brackets the 
population parameter either from above or below and furnishes 
an upper or lower bound to its magnitude. 

Confidence Interval for the Population Mean
-------------------------------------------

Assume samples are drawn from a normal population.

CI = sample statistic +/- Margin of error

Notes:

-  The sample statistic is the midpoint of the CI.

-  The SAMPLING ERROR is the half the width of the CI 

                   _  ^
                   x, p
 +------------------+-------------------+
LCI                                    UCI
 <------- SE ------><------- SE ------><

Confidence Interval for the Population Mean
-------------------------------------------

n ≥ 30:
          _                  _                                                           
σ known:  x - Zα/2σ/√n < μ < x + Zα/2σ/√n  
            _
σ unknown:  x - Zα/2s/√n < μ < x + Zα/2s/√n         

n ≤ 30:
          _                     _                                          
σ known:  x - tα/2,n-1σ/√n < μ < x + tα/2,n-1σ/√n  
            _                     _  
σ unknown:  x - tα/2,n-1s/√n < μ < x + tα,n-1/2s/√n         
                          _
In both cases s2 = Σ(xi - x)2/(n - 1)

Example: 95% CI Normal distribution
_                                                                     
x +/- 1.96σ/√n 

Confidence Interval for the Population Proportion
-------------------------------------------------

Again, assume samples are drawn from a normal population.  
This means that np and n(1 - p> ≥ 10.

^                    ^                                                      
p - Zα/2√(pq/n) ≤ p ≤ p + Zα/2√(pq/n)  where q = (1 - p)  

If n is not large enough the above procedure performs poorly
if p is small (i.e. np ≤ 10).  Under these circumstances
it is necessary to use Wilson's adjustment for p.  This is:

^
p = (x + 2)/(n + 4) 

Where x is the number of 'successes' in the sample.  The CI 
then becomes:

^                          ^                                                      
p - Zα/2√(pq/(n + 4)) ≤ p ≤ p + Zα/2√(pq/(n + 4))  

Rule of Thumb for Calculating σ
-------------------------------

σ ~ range/4 = (UCL - LCL)/4

Confidence Interval for the Population Variance and SD
------------------------------------------------------

Again, assume samples are drawn from a normal population.



P(χ21-α/2 < (n - 1)s2/σ2 < χ2α/2) = 1 - α

This leads to:

P((n - 1)s2/χ2α/2 < σ2 < (n - 1)s2/χ21-α/2) = 1 - α

Therefore, the CI is:

((n - 1)s2/χ2α/2,(n - 1)s2/χ21-α/2) 

and for the SD:

(√{(n- 1)s2/χ2α/2},√{(n- 1)s2/χ21-α/2}) 

Unlike the Student-t and Normal distribution, the χ2 distribution 
is asymmetric so the left and right tails are different.

Confidence Interval for the Ratio of the Population Variances
-------------------------------------------------------------

Again, assume samples are independent and drawn from a 
normal population.



P(F1-α/2 ≤ (s12/σ12)/(s22/σ22) ≤ Fα/2) = 1 - α

This leads to:

P((s12/s22)/Fα/2 ≤ σ12/σ22 ≤ (s12/s22)/F1-α/2) = 1 - α

Therefore, the CI is:

((s12/s22)/Fα/2,(s12/s22)/F1-α/2)

Like the χ2 distribution the F-distribution is asymmetric so 
the left and right tails are different.

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