The following methods and formulas are used for testing the ratio between the test mean and the reference mean.

Term | Description |
---|---|

ρ | Ratio |

Test mean | |

Reference mean |

The mean of the test sample, , is given by:

The mean of the reference sample, , is given by:

The standard deviation of the test sample, *S*_{1}, is given by:

The standard deviation of the reference sample, *S*_{2}, is given by:

Term | Description |
---|---|

X _{i} | Observations from the test sample, with i = 1, ..., n_{1} |

Y _{i} | Observations from the reference sample, with i = 1, ..., n_{2} |

n_{1} | Number of observations in the test sample |

n_{2} | Number of observations in the reference sample |

Let *k*_{1} be the value that you specify for the lower limit and *k*_{2} be the value that you specify for the upper limit. By default, the lower equivalence limit, *δ*_{1}, is given by:

and the upper equivalence limit, *δ*_{2}, is given by:

By default, the degrees of freedom for the test, *v*, are given by the following formula:

Minitab displays *v* rounded down to the nearest integer.

If you select the option to assume equal variances, then Minitab calculates the degrees of freedom as follows:

Term | Description |
---|---|

S_{1} | Standard deviation of the test sample |

n_{1} | Number of observations in the test sample |

S_{2} | Standard deviation of the reference sample |

n_{2} | Number of observations in the reference sample |

Term | Description |
---|---|

S_{p} | Pooled standard deviation |

S_{1} | Standard deviation of the test sample |

n_{1} | Number of observations in the test sample |

S_{2} | Standard deviation of the reference sample |

n_{2} | Number of observations in the reference sample |

Minitab cannot calculate the confidence interval (CI) if any of the following three conditions are not met:

- 100(1 - α)% CI
By default, Minitab calculates the 100(1 - α)% CI for ρ as follows:

CI = [min(C, ρ

where:_{L}), max(C, ρ_{U})] - 100(1 - 2α)% CI
If you select the option to use the 100(1 - 2α)% CI, then the CI is given by the following:

CI = [ρ_{L}, ρ_{U}]

If you select the option to assume equal variances, then the CI is calculated as follows.

Minitab cannot calculate the CI if any of the following three conditions are not met:

- 100(1 -α)% CI
Minitab calculates the 100(1 - α)% CI as follows:

CI = [min(C, ρ

Where:_{L}, max(C, ρ_{U})] - 100(1 - 2 α)% CI
If you select the option to use the 100(1 - 2 α)% CI, then the CI is given by the following:

CI = (ρ_{L}, ρ_{U})

For a hypothesis of Test mean / reference mean > lower limit, the 100(1 - α)% lower bound is equal to ρ_{L}.

For a hypothesis of Test mean / reference mean < upper limit, the 100(1 - α)% upper bound is equal to ρ_{U}.

Term | Description |
---|---|

Mean of the test sample | |

Mean of the reference sample | |

S_{1} | Standard deviation of the test sample |

n_{1} | Number of observations in the test sample |

S_{2} | Standard deviation of the reference sample |

n_{2} | Number of observations in the reference sample |

δ_{1} | Lower equivalence limit |

δ_{2} | Upper equivalence limit |

S_{ρ} | Pooled standard deviation |

v | Degrees of freedom |

α | Significance level for the test |

t_{1-α,v} | Upper 1 - α critical value for a t-distribution with v degrees of freedom |

Term | Description |
---|---|

Mean of the test sample | |

Mean of the reference sample | |

S_{1} | Standard deviation of the test sample |

n_{1} | Number of observations in the test sample |

S_{2} | Standard deviation of the reference sample |

n_{2} | Number of observations in the reference sample |

S_{ρ} | Pooled standard deviation |

δ_{1} | Lower equivalence limit |

δ_{2} | Upper equivalence limit |

The probability, *P*_{H0}, for each null hypothesis is given by the following:

If , then:

H_{0} |
P-Value |
---|---|

Term | Description |
---|---|

Λ | Unknown ratio of the mean of the test population to the mean of the reference population |

δ_{1} | Lower equivalence limit |

δ_{2} | Upper equivalence limit |

v | Degrees of freedom |

T | t-distribution with v degrees of freedom |

t_{1} | t-value for the hypothesis |

t_{2} | t-value for the hypothesis |

For information on how the t-values are calculated, see the section on t-values.