Wolfram Alpha:
Search by keyword:
Astronomy
Chemistry
Classical Mechanics
Classical Physics
Climate Change
Cosmology
Finance and Accounting
Game Theory
General Relativity
Group Theory
Lagrangian and Hamiltonian Mechanics
Macroeconomics
Mathematics
Mathjax
Microeconomics
Nuclear Physics
Particle Physics
Probability and Statistics
Programming and Computer Science
Quantitative Methods for Business
Quantum Computing
Quantum Field Theory
Quantum Mechanics
Semiconductor Reliability
Solid State Electronics
Special Relativity
Statistical Mechanics
String Theory
Superconductivity
Supersymmetry (SUSY) and Grand Unified Theory (GUT)
The Standard Model
Topology
Units, Constants and Useful Formulas
Hypothesis Testing
------------------
Hypothesis t esting asks the question is the sample statistic
different to the population parameter.
Null hypothesis H_{0}: =
Alternate hypothesis H_{1}: ≠ <= 2 tailed
< <= 1 tailed
> <= 1 tailed
So what are the 4 possible outcomes?
1. Correctly accept H_{0} and reject H_{1}
2. Correctly accept H_{1} and reject H_{0}
3. Accept H_{1} when H_{0} is true - TYPE 1 error = α
Where α is the SIGNIFICANCE LEVEL of the test.
4. Accept H_{0} when H_{1} is true - TYPE 2 error = β
Where 1 - β is the POWER of the test.
p-value
-------
The p-value for any hypothesis test is the α level at which
we would be indifferent between accepting or rejecting H_{0}.
That is, the p-value is the α level at which the given value
of the sample statistic is on the borderline between the
acceptance and rejection regions.
The p-value corresponds to the shaded areas. The 1-tailed
test is twice the 2-tailed value. Consider a Z-distribution
and α = 0.05:
1-tailed: p = 0.05 for a critical value of Z = -1.645 or +1.645
Therefore, if the computed Z score was < -1.645 we would reject
_{ } _
H_{0} and accept H_{1}: x < μ
If the computed Z score was < +1.645 we would reject
_{ } _
H_{0} and accept H_{1}: x > μ
2-tailed: p = 0.025 for a critical value of Z = -1.96 or +1.96
Therefore, if the computed Z score was < -1.96 or > +1.96 we
_{ } _
would reject H_{0} and accept H_{1}: x ≠ μ
The χ^{2} and F distributions are not symmetric
and so the left and right tails are different.
The p-value can also be thought of as the probability of
obtaining a test statistic as extreme as or more extreme
than the actual test statistic obtained, given that H_{0} is
true.
The null hypothesis is rejected if the p-value is less than
or equal to α (i.e. the test statistic falls with in the
shaded areas). In the language of statistics we say:
"At the α level of significance we can accept/reject H_{0},
there is not/is a difference (≠, < or >) between the sample
and the population".
F Test example:
Suppose you randomly select 7 marbles from company As
production line and 12 marbles from company Bs production
line and measure their diameters. Assume you are given:
s_{A} = 1.0 and s_{B} = 1.1
F = s_{A}^{2}/s_{B}^{2} = 1/1.21 = 0.83
H_{0}: σ_{A} = σ_{B}
H_{1}: σ_{A} ≠ σ_{B}
From tables F_{0.05} for v_{1} = n_{1} - 1 = 6 and v_{2} = n_{2} - 1 = 11 is equal
to 3.0946. Since 0.83 < 3.0946 the result is not significant and
there is no reason to reject H_{0}.