TECHNOLOGY OVERVIEW
# Hypothesis testing for the difference in the averages from two normally distributed population

Hypothesis testing for the difference in two averages

- Randomly sample n1 items from population 1 and n2 from population 2. N1 can be equal to n2.
- Calculate standard deviations and averages of both samples.
- Specify a level of significance
- Specify an equality (<. > or <>)
- Specify a reference value for the difference in averages.
- Determine the test statistic with the following expressions, depending on whether standard deviations are known/unknown and equal/not equal
- Both standard deviations are known
- Test statistic= (xbar1-xbar2)/sqrt(sigma1^2/n1+sigma2^/n2)
- Where xbar1 and xbar2 is equal to the sample averages sampled from population 1 and population 2
- Sigma1 and sigma2 are the known standard deviations for population 1 and population 2
- N1 and n2 are the respective sample sizes

- Both standard deviations are unknown but equal
- Test statistic= (xbar1-xbar2)/(pooled standard deviation*sqrt(1/n1+1/n2) on n1+n2-2 degrees of freedom.
- Where xbar1 and xbar2 is equal to the sample averages sampled from population 1 and population 2
- Pooled standard deviation=((n1-1)*Sd1^2+(n2-1)*Sd2^2)/(n1+n2-2)
- Sd1 and Sd2 are the sample standard deviations for population 1 and population 2
- N1 and n2 are the respective sample sizes

- Both standard deviations are unknown and not equal
- Test statistic= (xbar1-xbar2)/sqrt(sd1^2/n1+sd2^/n2)
- On
- [((Sd1^2/n1)/(Sd1^2/n1+Sd2^2/n2))^2/(n1-1)+ ((Sd2^2/n2)/(Sd1^2/n1+Sd2^2/n2))^2/(n2-1)]-^-1
- Degrees of freedom
- Where xbar1 and xbar2 is equal to the sample averages sampled from population 1 and population 2. Sd1 and Sd2 are the sample standard deviations for population 1 and population 2

- Both standard deviations are known
- Determine the critical region for the test statistics. The critical region depends on the equality i.e. < or > or <>.
- Compare the test statistic with the Critical region and conclude significance if the test statistic falls outside the critical region.

Alternatively, a p-value can be calculated and if the p-value falls below the specified level of significance the Null Hypothesis is rejected and the Alternative Hypothesis is accepted

BIS.Net Analyst and the BIS.Net Inferences APP both use p-values for hypothesis testing as these are more flexible than imposing a predefined level of significance on users. Both applications test for equality in variances automatically. However, this test is based on the F Test which is not robust to non-normality.