Hi
On 4 Dec 2003, VOLTOLINI wrote:
> The P is the probability of getting a value of a test statistic equal to or
> greater than the one observed in your data, if it is a random pick from the
> distribution of the test statistic that would be created if a specified null
> hypothesis were true. So, in the situation where the means of two
> distributions are being compared, a t statistic might be calculated. If the
> null hypothesis is that the distributions have an identical mean, the t
> statistic comes from a distribution of t with mean of zero. In such a
> distribution, there is a particular probability (P) that any given value of
> t will be equaled or exceeded if you pick t values at random. It is not to
> be confused with the probability you choose to decide to reject the null
> hypothesis (i.e. alpha, which is then the probability of rejecting the null
> hypothesis incorrectly - you think it is wrong when it is true, i.e. the
> probability of Type I error).
I would not so sharply distinguish between these two cases in
teaching hypothesis testing. If H0 is true, there is a certain
probability that t is >= any specified value of t (say t_a, where
a stands for the area above t_a). That is, p(t >= t_a IF H0
true) = a, where 0<=a<=1. How this general idea is best
communicated will depend on what students know or can be taught
about probability distributions in general, what is available in
the way of physical or computer simulations, time available,
sophistication of students, ....
A good foundation for the general idea of the probability
(sampling) distribution of t, will help students to realize that
the observed p and critical t approaches are equivalent and
arrive at identical conclusions if alpha is the same. For the
critical value approach, substitute t_critical for t_a and alpha
(say .05) for a to get: p(t >= t_critical IF H0 true) = .05.
Reject H0 if t_observed >= t_critical because the probability of
this outcome is < .05. For the observed p approach, substitute
t_observed for t_critical and p_observed for a to get: p(t >=
t_observed IF H0 true) = p_observed. If p_observed < .05
(alpha), then reject H0 because probability of this outcome is
< .05.
Note that
when t_observed is less than t_critical, then p_observed
will be greater than .05 (alpha),
when t_observed equals t_critical, p_observed will be
exactly .05, and
when t_observed is greater than t_critical, then
p_observed will be less than .05.
Best wishes
Jim
============================================================================
James M. Clark (204) 786-9757
Department of Psychology (204) 774-4134 Fax
University of Winnipeg 4L05D
Winnipeg, Manitoba R3B 2E9 [EMAIL PROTECTED]
CANADA http://www.uwinnipeg.ca/~clark
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