Select the method or formula of your choice.

The mean for the Poisson distribution is estimated as:

Data | 2 2 3 3 2 4 4 2 1 1 1 4 4 3 0 4 3 2 3 3 4 1 3 1 4 3 2 2 1 2 0 2 3 2 3 |

Category (i) |
Observed (O)_{i} |
Estimated mean | Poisson probability (p)_{i} |
---|---|---|---|

0 | 2 | 0 * 2 = 0 | p = _{0}e^{-2.4} = 0.090718 |

1 | 6 | 1 * 6 = 6 | p = _{1}e^{-2.4} * 2.4 = 0.217723 |

2 | 10 | 2 * 10 = 20 | p = _{2}e^{-2.4} * (2.4)^{2}/ 2! = 0.261268 |

3 | 10 | 3 * 10 = 30 | p = _{3}e^{-2.4} * (2.4)^{3}/ 3! = 0.209014 |

7 | 4 * 7 = 28 | p = 1 - (_{4}p + _{0}p +_{1}p + _{2}p) = 0.221267_{3} |

N = 35

Σ (*i* * *O _{i}*) = 84

Estimated Mean =

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

N | sum of all observed values (O_{0} + O_{1} + ...+ O)_{k} |

k | (the number of categories) - 1 |

O_{i} | the observed number of events in the i^{th} category |

p_{i} | Poisson probability |

Minitab determines categories using the following iterative methods:

Let *p _{i} * = P(

Let *i* = 1: if *N***p _{i} * 2, then the first category is defined as "

Conceptually, defining the last category is similar to defining the first category, but Minitab works backward starting from the largest data value.

The last category is " *x _{j} *", where

After determining the first and the last categories, Minitab determines the categories between them. Let "*X* *k*" be the first category, and "*X* *m*" is the last category. If all integers between (*k*, *m*) have expected values 2, then they all constitute a middle category. If not, Minitab uses a recursive loop to group multiple adjacent integers into categories with expected values 2. There are some situations, such as a data set with few observations, where the expected value of a category will be less than 2.

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

N | the total number of observations |

x _{i} | the i ^{th} value in the data set after sorting it from smallest to largest |

p _{i} | Poisson probability |

The Poisson probability of the *i* ^{th} category (*i* < *k*) is,

The Poisson probability for the last category, where *i* = *k*,

*p*_{i } = 1 – (*p*_{0} + *p*_{1} + ...+ *p*_{k-1})

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

k | the number of categories |

λ | the estimated mean from your sample |

The expected number of observations in *i* ^{th} category is *N* * *p _{i} *.

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

N | sample size |

p _{i} | the Poisson probability associated with the i ^{th }category |

Contribution of the *I*^{th }category to the chi-square value is calculated as

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

O _{I} | the observed number of observations in the I^{th} category |

E _{I} | the expected number of observations in the I^{th }category |

The chi-square goodness-of-fit test statistic is calculated as,

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

k | (the number of categories) - 1 |

O _{i} | the observed number of observations in the i^{th} category |

E _{i} | the expected number of observations in the i^{th }category |

The p-value is:

Prob (*X* > Test statistic)

where *X* follows a chi-square distribution with *k* - 1 degrees of freedom if you use the MEAN subcommand, or *k*- 2 degrees of freedom if you do not use the MEAN subcommand.

Data | 2 2 3 3 2 4 4 2 1 1 1 4 4 3 0 4 3 2 3 3 4 1 3 1 4 3 2 2 1 2 0 2 3 2 3 |

Category (i) |
Observed (O)_{i} |
Estimated mean | Poisson probability (p)_{i} |
---|---|---|---|

0 | 2 | 0 * 2 = 0 | p = _{0} e ^{-2.4} = 0.090718 |

1 | 6 | 1 * 6 = 6 | p = _{1} e ^{-2.4} * 2.4 = 0.217723 |

2 | 10 | 2 * 10 = 20 | p = _{2} e ^{-2.4} * (2.4)^{2}/ 2! = 0.261268 |

3 | 10 | 3 * 10 = 30 | p = _{3} e ^{-2.4} * (2.4)^{3}/ 3! = 0.209014 |

7 | 4 * 7 = 28 | p = 1 - (_{4} p + _{0} p +_{1} p + _{2} p) = 0.221267_{3} |

= ( 0.434920 + 0.344527 + 0.080058 + 0.985114 + 0.071545) = 1.91622

*k* = 5= the number of categories

DF = 5- 2 = 3

p-value = P (*X* > 1.91622) = 0.590

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

k | the number of categories |

O _{i} | the observed number of observations in the i^{th} category. |

E _{i} | the expected number of observations in the i^{th }category. |

chi-square goodness-of-fit test statistic | |

DF | degrees of freedom |