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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}.