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

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

D | Difference |

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 |

By default, Minitab uses the following formula to calculate the standard error of the difference, SE:

If you select the option to assume equal variances, then Minitab calculates the pooled standard deviation, S_{p}, and the standard error of the difference, SE, using the following formulas:

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 |

S_{p} | Pooled standard deviation |

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 |

By default, Minitab uses the following formula to calculate the 100(1 – α)% confidence interval (CI) for equivalence:

CI = [min(*C, D _{l}*), max(

where:

If you select the option to use the 100(1 – 2α)% CI, then the CI is given by the following formula:

CI = [*D _{l}, D_{u}*]

For a hypotheses of Test mean > reference mean or Test mean - reference mean > lower limit, the 100(1 – α)% lower bound is equal to *D _{L}*.

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

D | Difference between the test mean and the reference mean |

SE | Standard error |

δ_{1} | Lower equivalence limit |

δ_{2} | Upper equivalence limit |

v | Degrees of freedom |

α | The 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 difference between the mean of the test population and the mean of the reference population. By default, the t-values are calculated as follows:

For a hypothesis of Test mean > reference mean, *δ*_{1} = 0.

For a hypothesis of Test mean < reference mean, *δ *_{2} = 0.

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

D | Difference between the sample test mean and the sample reference mean |

SE | Standard error of the difference |

δ_{1} | Lower equivalence limit |

δ_{2} | Upper equivalence limit |

The probability, *P*_{H0}, for each null hypothesis (H_{0}) is given by the following:

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

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

Unknown difference between the mean of the test population and 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.