Wolfram Alpha:
Search by keyword:
Astronomy
Chemistry
Classical Physics
Climate Change
Cosmology
Finance and Accounting
General Relativity
Lagrangian and Hamiltonian Mechanics
Macroeconomics
Mathematics
Microeconomics
Particle Physics
Probability and Statistics
Programming and Computer Science
Quantum Field Theory
Quantum Mechanics
Semiconductor Reliability
Solid State Electronics
Special Relativity
Statistical Mechanics
String Theory
Superconductivity
Supersymmetry (SUSY) and Grand Unified Theory (GUT)
test
The Standard Model
Topology
Units, Constants and Useful Formulas
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.