# student t-test ```Review of Student’s t-test
Comparison of means between two populations
Assumptions
These assumptions allow the sampling distribution of the test statistic to be calculated.



normal distributions
equal variances
random sampling
Hypotheses

H0:
B   A  0

HA:
B   A  0
Test statistic
based on sample estimators of population means. The sampling distributions are:

2 2 
xB  x A ~ N   B   A , A  B 
nA nB 

z
x B  x A   B   A 
 A2
nA

 B2

nB
xB  x A
 1
1 
 2   
 n A nB 
~ N 0,1
under the null hypothesis, and assumption of equal valiances.
Substituting the sample estimator of the population variance:
xB  x A
t
 1
1 
s 2   
 n A nB 
~ tn
A
 nB  2
Decision rule
The decision on whether or not to reject the null hypothesis in favor of the alternative is based on the
observed value of the test statistic being far from its central value. The critical points are determined such
that the ex ante risk of false rejection (probability of ‘Type I’ error, denoted ‘’) is limited, traditionally
to levels such as 0.05 and 0.01. Thus the decision rule:

Reject H0 if
t  t1 2 
with degrees of freedom (‘dof’)
For example, if  = 0.05, then the critical points are
n A  nB  2 .
 t0.975  .
The null hypothesis is traditionally viewed as the status quo ante, the state of knowledge (or ignorance)
before new evidence is evaluated. Thus, limiting the risk of Type I error makes the hypothesis test a
conservative procedure. At the same time, the failure of some particular sample evidence to lead to the
rejection of the null hypothesis is not to be viewed as a confirmation of the null hypothesis. The test may
simply lack sufficient power to detect a meaningful violation.
P-value
Since the willingness to take risks is a subjective, but necessary, component of statistical testing, a
transformation of the test statistic is used that allows other subjective determinations of risk. The p-value
(‘probability value’) of the test is the probability of observing a value of the test statistic more damaging
to the null hypothesis than the value actually observed, or the ex post risk of rejecting the null hypothesis.
p - value  prob t  tobs 
With this transformation of the test statistic, the rejection rule can be restated as:

Reject H0 if
p 
```