In order to complete each significance test between traditional valuation methods and using real options, two steps were made:

– The first step consists of completing a variance equality test or F-Test on Excel in order to test the significance of a difference between two standard deviations. Two samples with sizes n_P and n_Q, coming from large populations, have standard respective variances and Let and be the respective variances of the two total populations and the accepted principle:

[A23.1]

where F corresponds to the Fisher–Snedecor law. Let us test the hypothesis Thus:

[A23.2]

and, if then:

[A23.3]

From these samples, we calculate The tabulation of the inverse spread function of the Fisher–Snedecor law allows us to get the real t such that P[T < t] = 95%.

Therefore, if we have a 95% chance of being right (or a 5% chance of being wrong), maintaining that, at the scale of entire populations, the standard deviations and are equal.

• – The second step depends on the result of the first. Let us say:
• = the square of the standard deviation of premiums observed in the first sample;
• = the square of the standard deviation of premiums observed in the second sample;
• m_P = the average of premiums over the entirety of operations from which the first sample is taken;
• m_q = the variance of premiums over the entirety of operations from which the second sample is taken;
• = average of premiums from the first sample;
• = average of premiums from the second sample;
• n_P = size of the first sample;
• n_Q = size of the second sample.
• - In the hypothesis where the two standard deviations are equal, the comparison test of the averages of two samples is characterized by the property by which variable T follows a Student law at n_P + n_Q − 2 degrees of freedom (Student test). That is:

[A23.4]

Thus, following the hypothesis m_P = m_Q, follows a Student law at n_P + n_Q – 2 degrees of freedom, we calculate for the samples:

[A23.5]

Furthermore, it is possible to determine the value of the real c such that:

[A23.6]

where T is a Student variable. In this case:

[A23.7]

where F is the distribution function of the Student law. Thus, for a level of confidence of 95%, α = 5% and obtained from the Student law table. Consequently, if – c < t₀ < c, the equality hypothesis of means can be accepted with an error risk of 5%. Inversely, if t₀ > c or if t₀ < – c, the averages can be considered as significantly different with an error risk of 5%.

• - If the unknown standard deviations are different, it is possible to use the Aspin–Welch test to find out if the averages of samples chosen are significantly different. In this case, following the equality hypothesis of means, the property for which variable obeys the Fisher–Snedecor law with the parameters n_P – 1 and n_Q – 1, supposing that:
• = the variance of the variable from the first sample;
• = the variance of the variable from the second sample;
• = the standard deviation of the variable for companies in the first sample;
• = the standard deviation of the variable for companies in the second sample;
• n_P = size of the first sample;
• n_Q = size of the second sample.
• - And:

The number t₀ that satisfies can be determined using the samples. The Fisher–Snedecor table gives the value of c such that P[T > c] = 5%, where T obeys the law F(m,n). Thus, if t₀ > c, the equality hypothesis of standard deviations must be rejected with a 5% chance of being wrong. Let us say:

[A23.8]

which obeys a Student law with n_P + n_Q – 2 degrees of freedom. Thus, supposing m_P = m_Q,

[A23.9]

obeys a Student law with n_P + n_Q – 2 degrees of freedom. The number

[A23.10]

can be calculated through testing. It is also possible to calculate the value of c such that P[- c < T < c ] = 1 – α, hence: where F is the distribution function for the Student law. Thus, with a 95% chance of being right, α = 5% and c = F⁻¹(0.975), which can be obtained directly thanks to the Student law table. Consequently, if - c < t₀ < c, the equality hypothesis of means can be accepted with a 5% chance of error. If the standard deviations are different, an Aspin–Welch test must be done. Let us say , which obeys the Student law with n degrees of freedom, where:

[A23.11]

The rule for the decision is the same as that of the Student test.