Rejection Region and Critical Values
We have seen that we reject H0 when p < ,
(where p is the probability of our results or more extreme
results, assumig H0 to be true).
Our p value is gotten by taking our test statistic (e.g. Z Y ) and
using its probability distribution (e.g. normal).
However, there is another way to see if we should reject H0.
For one tail (directional) tests
Recall that we have defined Z as the Z value such that the
probability (Z > Z) = .
e.g. Z.05 = 1.645; Z.01 = 2.33, etc.
Thus, whenever Z Y > Z, we know that p <  and we can
therefore reject H0 with a one-tail test with significance level .
For two tail (non-directional) tests
Z/2 is the Z value such that the probability (Z > Z/2) = /2.
e.g. if  = .05, /2 = .025 and Z/2 = 1.96
if  = .01, /2 = .005 and Z/2 = 2.575
Thus, whenever Z Y > Z/2, we know that p <  and we can
therefore reject H0 with a two-tail test with significance level .
Confidence Intervals and hypothesis testing
A. Suppose our null hypothesis is that Y = 0.
B.
If 0 is outside the 1 -  CI, we would reject that null
hypothesis, if using a two tailed test with significance
level .
C.
If 0 is within the 1 -  CI, we would not reject that null
hypothesis, if using a two tailed test with significance
level .
But even if 0 is within the confidence interval, this does not
prove that = 0. Remember that 0 is not the only possible value
of  that will be within the confidence interval.
The t-distribution
Recall
ZY 
Y  0
Y
To compute our standard error we did not use the population
standard deviation (Y) but rather used the sample estimate (sY).
Gosset, known as Student, proved that when we do that the
normal distribution is not quite the right distribution to use in
computing the probabilities. He showed that the proper
distribution is the t- distribution.
The t-distribution is like the normal in also being a bell-curve
but
a) it is somewhat wider (has bigger tails)
b) its exact form depends on degrees of freedom)