# 10.4 - Minitab: One-Way ANOVA

10.4 - Minitab: One-Way ANOVAIn one research study, 20 young pigs are assigned at random among 4 experimental groups. Each group is fed a different diet. (This design is a completely randomized design.) The data are the pigs' weights in kg after being raised on these diets for 10 months. We wish to determine if there are any differences in mean pig weights for the 4 diets.

- \(H_0: \mu_1 = \mu_2 = \mu_3 = \mu_4\)
- \(H_a:\) Not all \(\mu\) are equal

Feed_1 | Feed_2 | Feed_3 | Feed_4 |
---|---|---|---|

60.8 | 68.3 | 102.6 | 87.9 |

57.1 | 67.7 | 102.2 | 84.7 |

65.0 | 74.0 | 100.5 | 83.2 |

58.7 | 66.3 | 97.5 | 85.8 |

61.8 | 69.9 | 98.9 | 90.3 |

Contained in the Minitab file: ANOVA_ex.mpx

Note that in this file the data were entered so that each group is in its own column. In other words, the responses are in a separate column for each factor level. In later examples, you will see that Minitab will also conduct a one-way ANOVA if the responses are all in one column with the factor codes in another column.

##
Minitab^{®}
– One-Way ANOVA

To perform an Analysis of Variance (ANOVA) test in Minitab:

- Open the Minitab file: ANOVA_ex.mpx
- From the menu bar, select
*Stat > ANOVA > One-Way*. - Click the drop-down menu and select
*'Responses are in a separate column for each factor level'*. - Enter the variables
*Feed_1*,*Feed_2*,*Feed_3*, and*Feed_4*to insert them into the*Responses*box. - Choose the
*Comparisons*button and check the box next to*Tukey*. Under*Results*also select*Tests*. *OK*and*OK*

The result should be the following output:

##### Method

Null hypothesis | All means are equal |
---|---|

Alternative hypothesis | At least one mean is different |

Significance level | \(\alpha=0.05\) |

*Equal variances were assumed for the analysis*

##### Factor Information

Factor | Levels | Values |
---|---|---|

Factor | 4 | Feed_1, Feed_2, Feed_3, Feed_4 |

##### Analysis of Variance

Source | DF | Adj SS | Adj MS | F-Value | P-Value |
---|---|---|---|---|---|

Factor | 3 | 4703.2 | 1567.73 | 206.72 | 0.000 |

Error | 16 | 121.3 | 7.58 | ||

Total | 19 | 4824.5 |

##### Model Summary

S | R-sq | R-sq(adj) | R-sq(pred) |
---|---|---|---|

2.75386 | 97.48% | 97.01% | 96.07% |

##### Means

Factor | N | Mean | StDev | 95% CI |
---|---|---|---|---|

Feed_1 | 5 | 60.68 | 3.03 | (58.07, 63.29) |

Feed_2 | 5 | 69.24 | 2.96 | (66.63, 71.85) |

Feed_3 | 5 | 100.340 | 2.164 | (97.729, 102.951) |

Feed_4 | 5 | 86.38 | 2.78 | (83.77, 88.99) |

*Pooled StDev = 2.75386*

##### Grouping Information Using the Tukey Method and 95% Confidence

Factor | N | Mean | Grouping | |||
---|---|---|---|---|---|---|

Feed_3 | 5 | 100.34 | A | |||

Feed_4 | 5 | 86.38 | B | |||

Feed_2 | 5 | 69.24 | C | |||

Feed_1 | 5 | 60.68 | D |

*Means that do not share a letter are significantly different.*

Difference of Levels |
Difference of Means |
SE of Difference |
95% CI | T-Value | Adjusted P-Value |
---|---|---|---|---|---|

Feed_2 - Feed_1 | 8.56 | 1.74 | (3.57, 13.55) | 4.91 | 0.001 |

Feed_3 - Feed_1 | 39.66 | 1.74 | (34.67, 44.65) | 22.77 | 0.000 |

Feed_4 - Feed_1 | 25.70 | 1.74 | (20.71, 30.69) | 14.76 | 0.000 |

Feed_3 - Feed_2 | 31.10 | 1.74 | (26.11, 36.09) | 17.86 | 0.000 |

Feed_4 - Feed_2 | 17.14 | 1.74 | (12.15, 22.13) | 9.84 | 0.000 |

Feed_4 - Feed_3 | -13.96 | 1.74 | (-18.95, -8.97) | -8.02 | 0.000 |

##### Tukey Simultaneous Tests for Differences of Means

*Individual confidence level = 98.87%*