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Next: Paired Data
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INFERENCES WITH TWO SAMPLES
- two random samples taken independently of each other:
- m obs. X1,X2,...,Xm from population 1 with mean m1 and S.D. s1.
- n obs. Y1,Y2,...,Yn from population 2 with mean m2 and S.D. s2.
- Inferences to be made on m1-m2.
Point Estimator for m1-m2 is 
:
- E[
]=m1-m2 - S.D.[] =
To test H0:m1-m2=D0 we use the same approach as we did with H0:m1=m0 but with the above mean and S.D. (instead of m and s/Ön...):
CASE 1: Normal populations with s1 and s2 known.
Test Statistic is z= 
Ha: m1-m2>D0Þ Reject if z³ za
Ha: m1-m2<D0Þ Reject if z£ -za
Ha: m1-m2¹D0Þ Reject if z³ za/2 OR z£ -za/2, i.e. |z|³za/2
A 100*(1-a)% confidence interval for m1-m2 is given by 
CASE 2: Arbitrary population with large sample sizes m and n.
Identical to above if s1 and s2 are known, otherwise we use
the estimates s1 and s2 in place of s1 and s2.
CASE 3: Normal populations with s1, s2 unknown and small samples.
Case 3a Suppose we can assume that s1 = s2 = s, i.e., means of the two populations could be
different but their variances are the
same.
Define the Pooled
Estimator of s2 as
Then the Test Statistic is t =
Ha:
m1-m2>D0 Þ Reject if t ³ ta,m+n-2
Ha:
m1-m2<D0 Þ Reject if t £ -ta,m+n-2
Ha:
m1-m2¹D0 Þ Reject
if t³ta/2,m+n-2 OR t£-ta/2,m+n-2,
i.e.
|t|³ta/2,m+n-2
A 100*(1-a)% confidence interval for m1-m2 is given by
Case 3b Suppose s1 ¹ s2, i.e., the variances of the
two populations are not the same. Then
T= 
has an approximate t distribution with degrees of freedom n estimated via 
rounded down to the nearest integer.
Therefore after computing the df= n via the above formula, the Test Statistic is t =
Ha:
m1-m2>D0 Þ Reject if t ³ ta,n
Ha:
m1-m2<D0 Þ Reject if t £ -ta,n
Ha:
m1-m2¹D0 Þ Reject
if t³ta/2,n OR t£-ta/2,n
i.e.
|t|³ta/2,n
A 100*(1-a)% confidence interval for m1-m2 is given by
In general the pooled procedure is better (lower Type
II error probability b for the same Type I error
probability a) if we are reasonably sure
that s1=s2. If not, it could lead to erroneous
conclusions and the two-sample t-test approach is preferable.
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Next: Paired Data
Previous: The P-Value for a test
Jayant Rajgopal