Page 393 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes (b) Another person examines the same data but with a preset value for α = .05. This
person is willing to support a higher risk of a Type I error, and hence the decision
≤
is to reject H because the p-value is less than α (.0222 .05) . It is important to
0
emphasize that the value of α used in the decision rule is preset and not selected
after calculating the p-value.
As we can see from Example 1, the level of significance represents the probability
of observing a sample outcome more contradictory to H than the observed sample
0
result. The smaller the value of this probability, the heavier the weight of the sample
evidence against H . For example, a statistical test with a level of significance of p =
0
.01 shows more evidence for the rejection of H than does another statistical test
0
with p = .20.
Example 2 : Using a preset value ofα = .05, is there sufficient evidence in the data to support the
research hypothesis ?
Solution : The null and alternative hypotheses are
H : μ ≥ 33
0
H: μ < 33
α
From the sample data, with s replacing σ , the computed value of the test statistic is
y − μ 0 31.2 − 33
z = = = – 1.27
σ n 8.4 35
The level of significance for this test statistic is computed by determining which values
of y are more extreme to H than the observed y . Because H specifies μ less than
0 α
33, the values of y that would be more extreme to H are those values less than 31.2,
0
the observed value. Thus,
p-value = P(y ≤ 31.2 , assuming μ = 33) = P(z ≤−1.27) = .1020
There is considerable evidence to support H . More precisely, p-value = .1020 > .05 =
0
α , and hence we fail to reject H . Thus, we conclude that there is insufficient evidence
0
(p-value = .1020) to support the research hypothesis. Note that this is exactly the same
conclusion reached using the traditional approach.
For two-tailed tests, H: μ ≠ μ , we still determine the level of significance by
α
0
computing the probability of obtaining a sample having a value of the test statistic
that is more contradictory to H than the observed value at the test statistic. However,
0
for two-tailed research hypotheses, we compute this probability in terms of the
magnitude of the distance from y to the null value of μ because both values of y
much less than μ and values of y much larger than μ contradict μ = μ . Thus,
0
0
0
the level of significance is written as
p-value = ( y − P μ ≥ 0 observed y − μ 0 ) = ( z ≥ computed ) z
P
= ( ≥2P z computedz )
To summarize, the level of significance (p-value) can be computed as

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