Equipe Raisonnement Induction Statistique
Consider an experiment involving two crossed factors Age and Treatment,
each with two modalities.
The means of the four experimental conditions (with 10 subjects in each) are respectively 5.77
(a1,t1), 5.25 (a2,t1), 4.83 (a1,t2) and 4.71 (a2,t2). 
It is strongly suggested to the reader that it has been demonstrated both a large main effect of treatment and a small interaction effect. 
Do you agree with these conclusions? 
There is nothing of the kind! 
The difference between the two observed treatment means is:

This clearly shows that it cannot be concluded both to a substantive difference between treatment means and to a small, or at least relatively negligible, interaction effect (and more again to a null interaction).. 
In an introductory statistical textbook, in a serie for the "grand public",
whose goal is to give the reader the possibility to "access the deep intuitions in the field",
one can find the following interpretation of a confidence interval for a proportion.

Do you agree with this interpretation? 
If you are not (again) a Bayasian and if your real intuition is that interpretation is, either right, or perhaps wrong but in any case desirable, you must seriously ask yourself if you are not a Bayesien "without knowing it". 
In the frequentist framework the possible values for the parameter cannot probabilised.
If, as in this example, the bounds computed for the observed sample are [0.58,0.64],
the event
"0.58<π<0.64"
is true or false
(because π is fixed), and we cannot give it a probability (other than 1 ou 0).

Ironically, it is the natural (Bayesian) interpretation of confidence
intervals in terms of "a fixed interval having a 95% chance of including
the true value of interest" which is their appealing feature.

I have find an article that report the results of a study designed to test the efficacy of a drug
by comparing two groups (treatment vs placebo) of 15 patients each.
The gives the observed difference d=+1.52 in favour of the treatment,
and a "Student t test": t=+0.683, 28 degrees of freedom, p=0.50, nonsignificant.

Is it possible?

Yes! 
For a 100(1α)%
interval, it is sufficient to know
t{(1α)/2}:
the (1α)/2 upper
point of the Student distribution with q degrees of freedom. [ d  (d/t)t{(1α)/2} , d + (d/t)t{(1α)/2} ] We find here for α = 0.05 and q=28 degrees of freedom t{0.975}= +2.0484, hence the 95% interval (of course it is assumed that d and t are computed with the needed accuracy):
[3.04,+6.08]

This interval can be interpreted as a 95% "frequentist" confidence interval or as a 95% "fiducialBayesian" interval. 
Consider the results of a study designed to test the efficacy of a drug by comparing two groups
(treatment vs placebo) of 15 patients each.

What conclusion would you draw for the efficacy of the drug? 
Answer spontaneously (without computation) 
From a normative viewpoint, the task involves the following simple and general result: d ± 2(d/t) hence here [2.93,+5.97] This very simple approximation is generally sufficient (the exact interval is [3.04,+6.08]). This straightforward and easily interpretable result should theoretically prevent the abusive interpretation of a nonsignificant result as "proof of the null hypothesis". Clearly here the data cannot lead to conclude to the inefficacy of the drug (because of the great variability observed. 
However, in front of this situation 84% of the professional applied statisticians and 85% of
the psychologists (all with experience in processing and analyzing experimental data) concluded inefficacy.

In a study that compares an experimental condition to a control condition, a difference +1.82 between the two moyennes has been observed. The difference is significant at two tailed level 0.05: t=+2.09, 19 degrees of freedom, p=0.05. 
(1) What, for you, is the probability that, in a replication of the experiment,
the observed difference will be positive? 
Answer spontaneously (without computation) 
From a normative viewpoint, since there is no a priori information external to the experiment,
it seems reasonable to base the prediction on the data only. 
The majority of investigated psychological researchers underestimated the first probability
and overestimated the second probability.

A pair of socks is (blindly) draw from a drawer in which there are a pair of red socks and a pair
of green socks.

Do you think there is:

Answer spontaneously (without computation) 
The correct response is:
2) more chance of obtaining result 2

If you answered "an equal chance of obtaining the two results" (equiprobability bias),
you are part of the majority.

Consider the two following events:
"The fact that a pair of socks that match is obtained from a blindly draw of two socks from a drawer in which
there are two pairs of different socks"

Do you think that randomness is involved or not in each of these two events? 
Answer spontaneously 
Of course, there is no "good response"! 
Three groups of subjects have been questioned: college undergraduates students,
researchers in psychology, and researchers in mathematics and statistics.
