May 2, 2009 - Class Notes

A. Testing the Difference Between Means (Large Independent Samples)

A null hypothesis, H0, is a statistical hypothesis that usually states that there is no
difference between the parameters of two populations. The null hypothesis always contains
the symbol ≤, =, or ≥.

An alternative hypothesis, Ha, is a statistical hypothesis that is true when H0 is false. The
alternative hypothesis always contains the symbol >, ≠, or <.

The Central Limit Theorem states that the difference of the sample means is normally
distributed when the following conditions are satisfied:
• The samples are randomly selected.
• The samples are independent.
• Each sample size is at least 30, or, if not, each population has a normal distribution
with a known standard deviation σ.
These three conditions are often called the assumptions of the statistical test.
When the difference of sample means is normally distributed:

B. Testing the Difference Between Means (Small Independent Samples)
When small samples—n < 30—are used and the population standard deviation is unknown,
the Central Limit Theorem does not apply. In this case, you can use a t-test to test the
difference between two population means μ1 and μ2 if the following conditions are met:

• The samples are randomly selected.
• The samples are independent.
• Each population has a normal distribution.