The p-value

Sunday, 18 August 2019

12:51 AM

The p-value is the smallest level of significance that would lead to rejection of ﷐𝐻﷮0﷯ with the observed sample

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P - 2 X ( 1 - $ 120 1 ) )

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p = P (X [Ito ± IR — "01] when 110)
= 1 —P (X e [go ± IR — [tol] when "o)
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Define zo as the z-score of R if kto, i.e.
"observed value of the test statistic"
X-"0 JV(0 1), we have
As we know that Z
p=1-P n
= 1 —P(Z e [—141, Izol]) (1 —
(by symmetry of the M (0 1) distribution)

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7. Inferences concerning a mean
p-value: example
7.8 Tests of Hypotheses
In our example, we have observed a sample mean of R 20.75 min,
so that the p-value is given by
p - 1 -P(X e [19.25,20.75] when 20)
We have here that
20.75 - 20
36
2.25
2
so that the p-value can easily be computed:
Matlab
p -1 -P(-2.25 S Z S 2.25) = 2 x (1 - Ø(2.25))
0.024
If Ho 20 is true, the probability of obtaining another random sample
whose mean is at least as far from 20 as our 20.75 is 0.024.
if we rejected Ho, we would be wrong with a 2.4% chance
MATH2099/2859 (Statistics)
Dr Jia Deng
Term 2, 2019 - Lecture 7
24/53

Ink Drawings
Ink Drawings

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Operationally, once a p-value is computed, we typically compare it to a
predefined significance level to make a decision:
if p < a , reject Ho,
if p 2 a , do not reject Ho

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7. Inferences concerning a mean
7.8 Tests of Hypotheses
Procedures in hypothesis testing
O State the null and alternative hypotheses: Ho and Ha
O Determine the rejection criterion
O Compute the appropriate test statistic and determine its
distribution
O Calculate the p-value using the test statistics computed
O Conclusion: reject/do not reject Ho, relate back to the research
question
MATH2099/2859 (Statistics)
Dr Jia Deng
Term 2,
2019
- Lecture 7
26/53

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One-sided alternatives
O With Ha : > go, we are after a constant u such that
> u when
As we know that Z n M (0 1),
+ the decision rule is reject Ho if R > + Zl_a—

Ink Drawings

Note: For one sided, we only reject if it is MUCH less/greater than, so we use a z-value of 𝟏−𝜶 instead of 𝟏−﷐𝜶﷮𝟐﷯

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Normal populations with unknown standard deviation
For the one-sided alternatives, the rejection criteria are
s
reject Ho if R > + tn-l,l-a
n
and the associated p-values are
or
s
reject Ho if R < — tn-l,l-a
n
Note: For one sided, we only reject if it is MUCH less/greater than, so we use a z-value of 𝟏−𝜶 instead of 𝟏−﷐𝜶﷮𝟐﷯

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