Supplementary exercise 6.107 of IPS7e
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Let's say for simplicity of discussion that the test was for a population mean 
being equal to zero, i.e. H0: mu=0 (where mu is the population mean).

With P=0.95, there is absolutely no evidence against H0 (at the common
significance level of 0.05, or any other realistic significance level).
We can say the data were not at all surprising if H0 was true. Therefore
we have no reason (and certainly no justification) to question H0 from
our data: it could very well be true.

Now let us consider a couple of the misconceptions from the Greenland et
al. (2006) paper. 

Regarding 4.), we should not say that the null hypothesis is true or
should be accepted. That is simply because the test cannot give us
evidence in favour of the null hypothesis. Yes, H0 could be true, but
there are (infinitely) many other hypotheses that could be true as well. 
A 95% confidence interval gives us the range of values for which H0 would
not be rejected. Some of those values in the range will have a smaller 
P-value than the mu=0 from our H0, but also for them we could say the
observed data were not surprising. The bottom line is that we no proof
or evidence for the null hypothesis.

Regarding 5.), many of the same comments apply. With our typical two-sided
or one-sided interval alternative hypothesis, there are many values among
those included in the alternative for which a test of that particular mean
would give a higher P-value than the 0.95 we got for mu=0. So it just wrong
to think that a large P-value favours the H0 over the alternative. The
situation would be different if the alternative was also a single value, 
because then it comes simply a choice between two values, but this is not
the situation we have been considering in the course.

Regarding 6.), the second part of the statement was also already
discussed under 4.), and it is also wrong that no effect was observed.
We can really only say that no effect was observed if the estimated mean
is exactly zero. That would however correspond to a P-value of 1 (for
a two-sided alternative; for example, the z- or t-statistic would be equal 
to zero), but we had P=0.95. So the observed mean must have been
off zero, possibly slightly but still off, and therefore the observed 
effect was not zero.

Finally regarding 8.), if we again think about z- or t-statistics, then
a small value of these will occur either because the numerator (i.e.,
the estimate minus the hypothesized value) is small (and that would be 
a small effect size) or the denominator (i.e., the standard error of
the estimate) is large. From looking at the P-value alone there is 
no way we can exclude the second explanation. A confidence interval 
for the population mean will give us a plausible range of values 
of the population mean, and that will be our best way to represent 
the information we have about effect size from our data.