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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 - Z_{α/2}σ/√n < μ < x + Z_{α/2}σ/√n
_
σ unknown: x - Z_{α/2}s/√n < μ < x + Z_{α/2}s/√n
n ≤ 30:
__{ } _
σ known: x - t_{α/2,n-1}σ/√n < μ < x + t_{α/2,n-1}σ/√n
__{ } _
σ unknown: x - t_{α/2,n-1}s/√n < μ < x + t_{α,n-1/2}s/√n
^{ } _
In both cases s^{2} = Σ(x_{i} - 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 - Z_{α/2}√(pq/n) ≤ p ≤ p + Z_{α/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 - Z_{α/2}√(pq/(n + 4)) ≤ p ≤ p + Z_{α/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(χ^{2}_{1-α/2} < (n - 1)s^{2}/σ^{2} < χ^{2}_{α/2}) = 1 - α
This leads to:
P((n - 1)s^{2}/χ^{2}_{α/2} < σ^{2} < (n - 1)s^{2}/χ^{2}_{1-α/2}) = 1 - α
Therefore, the CI is:
((n - 1)s^{2}/χ^{2}_{α/2},(n - 1)s^{2}/χ^{2}_{1-α/2})
and for the SD:
(√{(n- 1)s^{2}/χ^{2}_{α/2}},√{(n- 1)s^{2}/χ^{2}_{1-α/2}})
Unlike the Student-t and Normal distribution, the χ^{2} 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(F_{1-α/2} ≤ (s_{1}^{2}/σ_{1}^{2})/(s_{2}^{2}/σ_{2}^{2}) ≤ F_{α/2}) = 1 - α
This leads to:
P((s_{1}^{2}/s_{2}^{2})/F_{α/2} ≤ σ_{1}^{2}/σ_{2}^{2} ≤ (s_{1}^{2}/s_{2}^{2})/F_{1-α/2}) = 1 - α
Therefore, the CI is:
((s_{1}^{2}/s_{2}^{2})/F_{α/2},(s_{1}^{2}/s_{2}^{2})/F_{1-α/2})
Like the χ^{2} distribution the F-distribution is asymmetric so
the left and right tails are different.