P-Values – Denoted p, this is the smallest
significance level at which the null hypothesis would
not be rejected. Sometimes referred to as the observed
level of significance.
Finding the P-Value
find the value in the appropriate (z or t) table that
corresponds to the calculated value of the test statistic
and
i)report this value if the test is one tailed and the results
are on the critical side of the hypothesized value
ii) report .5000+the value if the test is one tailed and the
results are on the non-critical side of the hypothesized
value
iii) reporting double the value if the hypothesis is two
tailed
Finding p-values – Lower tailed tests
f(z)
p-value
µ0
-z
0
P-Value is the area below the observed value of
the test statistic
x
z
Finding p-values – Upper tailed tests
f(z)
p-value
µ0
x
+z
0
P-Value is the area above the observed value of
the test statistic
z
Finding p-values – Two tailed tests
f(z)
1
2
p-value
µ0
+z
0
P-Value is two times the area more extreme than
the observed value of the test statistic
x
z
Finding p-values – Two tailed tests
f(z)
1
2
p-value
µ0
x
-z 0
z
P-Value is two times the area more extreme than
the observed value of the test statistic
How do you use p-values?
if the p value is less than the significance level α, we
reject the null hypothesis
Why are p-values preferred?
a. they allow anyone to select their own significance level α
b. they provide a measure of the strength of the evidence the
sample data provides against the null hypothesis (smaller
p-value – stronger evidence against H0)
c. they are usually reported by computer packages
Most software give two tailed p-values for the null
hypothesis that the parameter being tested is equal to zero
– what do you do when you want to test a one-tailed null
hypothesis (i.e., that the parameter being tested is no more
than zero or is no less than zero)?
If the sample value of the test statistic is in the same tail
as the reject region, i.e.
• the sample value of the test statistic exceeds the
hypothesized value in an upper tailed test or
• the sample value of the test statistic is less than the
hypothesized value in a lower tailed test
the p value is half the p-value reported by the software
If the sample value of the test statistic is in the opposite
tail of the reject region, i.e.
• the sample value of the test statistic exceeds the
hypothesized value in a lower tailed test or
• the sample value of the test statistic is less than the
hypothesized value in an upper tailed test
the p value is 1 – (half the p-value reported by the
software)