8.14 Testing The Difference Between Means
denote the means of the samples drawn from the first and second population respectively, having means m_{1} and m_{2} and standard deviations s_{1} and s_{2} and if the sizes of the samples are n_{1} and n_{2}, then it can be proved that the distribution of the difference between the means is normal with mean (m_{1}  m_{2} ) and S.D. is given by
Therefore,
Further, under the hypothesis H_{o} : m_{1} = m_{2} or H_{o} : m_{1}  m_{2} = 0 We see that
When the two samples belong to the same population, we have s_{1} = s_{2} = s then,
Similarly, the confidence limits for ( m_{1}  m_{2} ) at various levels of confidence are :
1)± 1.96 S at 95% level of confidence
2)± 2.58 S at 99% level of confidence
3)± 3. S at 99.73% level of confidence.
1. Note: Here S = S.E. =
For the samples drawn from the same population.
2. Note: If S.D. of two populations
i.e. s_{1} , s_{2}
are unknown, we use s.d. of samples in their places.
