Select the method or formula of your choice.

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

estimate of the first population proportion | |

estimate of the second population proportion | |

n _{1} | number of trials in the first sample |

n _{2} | number of trials in the second sample |

z _{α/2} | inverse cumulative probability of the standard normal distribution at 1 – α/2 |

α | 1 – confidence level/100 |

The calculation of the test statistic, *Z*, depends on the method used to estimate *p*.

- Separate estimates of p
- By default, Minitab uses separate estimates of p for each population and calculates Z as follows:
- Pooled estimate of p
- If the hypothesized test difference is zero and you choose to use a pooled estimate of
*p*for the test, Minitab calculates Z as follows:

The p-value for each alternative hypothesis is:

- H
_{1}:*p*_{1}>*p*_{2}: p-value = P(*Z*_{1}≥*z*) - H
_{1}:*p*_{1}<*p*_{2}: p-value = P(*Z*_{1}≤*z*) - H
_{1}:*p*_{1}≠*p*_{2}: p-value = 2P(*Z*_{1}≥*z*)

Calculate these probabilities on the standard normal distribution.

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

p_{1} | true proportion of events in the first population |

p_{2} | true proportion of events in the second population |

observed proportion of events in the first sample | |

observed proportion of events in the second sample | |

n_{1} | number of trials in the first sample |

n_{2} | number of trials in the second sample |

d_{0} | hypothesized difference between the first and second proportions |

x_{1} | number of events in the first sample |

x_{2} | number of events in the second sample |

Minitab performs Fisher's exact test in addition to a test based on a normal approximation. Fisher's exact test is valid for all sample sizes.

Under the null hypothesis, the number of events in the first sample (*x*_{1}) has a hypergeometric distribution with these parameters:

- Population size =
*n*_{1}+*n*_{2} - Number of events in population =
*x*_{1}+*x*_{2} - Sample size =
*n*_{1}

Let f( ) and F( ) denote the PDF and CDF of this hypergeometric distribution, respectively. Let Mode denote its mode. The p-values for each alternative hypothesis are as follows:

- H
_{1}:*p*_{1}<*p*_{2}p-value =

*F*(*x*_{1}) - H
_{1}:*p*_{1}>*p*_{2}p-value = 1 –

*F*(*x*_{1}– 1) - H
_{1}:*p*_{1}≠*p*_{2}Three cases exist:- Case 1:
*x*_{1}< Modep-value = p-lower + p-upperTerm Description p-lower *F*(*x*_{1})p-upper 1 – *F*(*y*– 1)*y*smallest integer > Mode such that *f*(*y*) <*f*(*x*_{1})###### Note

p-upper may equal zero.

- Case 2:
*x*_{1}= Modep-value = 1.0

- Case 3:
*x*_{1}> Modep-value = p-lower + p-upperTerm Description p-upper 1 – *F*(*x*_{1}– 1)p-lower *F*(*y*)y largest integer < Mode such that *f*(*y*) <*f*(*x*_{1})###### Note

p-lower may equal zero.

- Case 1:

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

p_{1} | true proportion of events in the first population |

p_{2} | true proportion of events in the second population |

x_{1} | number of events in the first sample |

x_{2} | number of events in the second sample |

n_{1} | number of trials in the first sample |

n_{2} | number of trials in the second sample |