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 |

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:

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

v | Degrees of freedom |

n | Number of pairs of observations |

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

X_{i} | The i^{th} observation in the test sample, such that (X_{i}, Y_{i}) is the i^{th} pair of observations |

Y_{i} | The i^{th} observation in the reference sample, such that ( X_{i}, Y_{i}) is the i^{th} pair of observations |

Mean of the test sample | |

Mean of the reference sample | |

n | Number of pairs of observations |

Minitab cannot calculate the confidence interval (CI) if either of the following two conditions are satisfied:

If the conditions are satisfied, Minitab calculates the CI based on the method used for the analysis.

- 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}] - One-sided intervals
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_{12} | Sample covariance between the X values and the Y values |

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

n | the sample size |

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

δ_{1} | Lower equivalence limit |

δ_{2} | Upper equivalence limit |

v | Degrees of freedom |

α | Significance level for the test (alpha) |

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

Let *t*_{1} be the t-value for the hypothesis, , and let *t*_{2} be the t-value for the hypothesis, , where is the ratio of the mean of the test population to the mean of the reference population.

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

Mean of the test sample | |

Mean of the reference sample | |

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

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

S_{12} | Correlation between the X values and the Y values |

n | Number of pairs of observations |

δ_{1} | Lower equivalence limit |

δ_{2} | Upper equivalence limit |

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

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.