Testing Hypothesis Concerning the equality of the Variances of Two Populations

Investment Analysis

Authors

• Narayana Rao

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Version 2

Last edited: 15 Nov 2008

Exported: 26 Nov 2011

Original URL: http://knol.google.com/k/-/-/2utb2lsm2k7a/490

 

To test if the variances of two populations are equal, or if the variance of one population is greater than (or less than) the variance of another population.  

             

 

The test statistic that is used to decide whether to reject these types of null or alternative hypotheses is the F-Statistic, which is defined as:

 

              F  =  S₁²/ S₂²

 

The F-distribution is a family of distributions that are defined by two separate degrees of freedom: the degrees of freedom in the numerator (n-1) and the d of f in the denominator (n-1).

 

Example: Perform a hypothesis test to determine at a 5% level of significance whether the manager’s standard deviations are equal.

 

                                           Manager A          Manager B

!0-year Avg. Annual Return   12%                          25%

Std. Deviation of Returns       20%                          30%

Variance                                 400                           900

 

Null    H0:   Sigma square of A  =  Sigma square of B

 

Alt      H1:   Sigma square of A  ≠  Sigma square of B

 

F calculated = S_A²/ S_B²    = 900/400 = 2.25

 

Note: By convention, the higher variance is put in the numerator when calculating the F-statistic.

 

The d of f of both numerator and denominator are 10-1 = 9.

 

This is a two tailed test and hence F(.025,9,9) is 4.03.

 

The acceptance range is 0 to 4.03. Since the calculated value of F is 2.25, it is inside the acceptance range. We cannot reject the null hypothesis.

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Narayana Rao - 15 Dec 2010