The following are facts about the *F* distribution:

- The curve is not symmetrical but skewed to the right.
- There is a different curve for each set of
*df*s. - The
*F*statistic is greater than or equal to zero. - As the degrees of freedom for the numerator and for the denominator get larger, the curve approximates the normal.
- Other uses for the
*F*distribution include comparing two variances and two-way analysis of variance. Two-way analysis is beyond the scope of this chapter.

### Example 13.2

Let’s return to the slicing tomato exercise in Try It. The means of the tomato yields under the five mulching conditions are represented by *μ*_{1}, *μ*_{2}, *μ*_{3}, *μ*_{4}, *μ*_{5}. We will conduct a hypothesis test to determine if all means are the same or at least one is different. Using a significance level of 5 percent, test the null hypothesis that there is no difference in mean yields among the five groups against the alternative hypothesis that at least one mean is different from the rest.

MRSA, or *Staphylococcus aureus*, can cause serious bacterial infections in hospital patients. Table 13.6 shows various colony counts from different patients who may or may not have MRSA. The data from the table is plotted in Figure 13.5.

Conc = 0.6 | Conc = 0.8 | Conc = 1.0 | Conc = 1.2 | Conc = 1.4 |
---|---|---|---|---|

9 | 16 | 22 | 30 | 27 |

66 | 93 | 147 | 199 | 168 |

98 | 82 | 120 | 148 | 132 |

Plot of the data for the different concentrations:

Test whether the mean numbers of colonies are the same or are different. Construct the ANOVA table by hand or by using a TI-83, 83+, or 84+ calculator, find the *p*-value, and state your conclusion. Use a 5 percent significance level.

### Example 13.3

Four sororities took a random sample of sisters regarding their grade means for the past term. The results are shown in Table 13.7.

Sorority 1 | Sorority 2 | Sorority 3 | Sorority 4 |
---|---|---|---|

2.17 | 2.63 | 2.63 | 3.79 |

1.85 | 1.77 | 3.78 | 3.45 |

2.83 | 3.25 | 4.00 | 3.08 |

1.69 | 1.86 | 2.55 | 2.26 |

3.33 | 2.21 | 2.45 | 3.18 |

Using a significance level of 1 percent, is there a difference in mean grades among the sororities?

Four sports teams took a random sample of players regarding their GPAs for the last year. The results are shown in Table 13.8.

Basketball | Baseball | Hockey | Lacrosse |
---|---|---|---|

3.6 | 2.1 | 4.0 | 2.0 |

2.9 | 2.6 | 2.0 | 3.6 |

2.5 | 3.9 | 2.6 | 3.9 |

3.3 | 3.1 | 3.2 | 2.7 |

3.8 | 3.4 | 3.2 | 2.5 |

Use a significance level of 5 percent and determine if there is a difference in GPA among the teams.

### Example 13.4

A fourth-grade class is studying the environment. One of the assignments is to grow bean plants in different soils. Tommy chose to grow his bean plants in soil found outside his classroom mixed with dryer lint. Tara chose to grow her bean plants in potting soil bought at the local nursery. Nick chose to grow his bean plants in soil from his mother’s garden. No chemicals were used on the plants, only water. They were grown inside the classroom next to a large window. Each child grew five plants. At the end of the growing period, each plant was measured, producing the data in inches in Table 13.9.

Tommy’s Plants | Tara’s Plants | Nick’s Plants |
---|---|---|

24 | 25 | 23 |

21 | 31 | 27 |

23 | 23 | 22 |

30 | 20 | 30 |

23 | 28 | 20 |

Does it appear that the three soils in which the bean plants were grown produce the same mean height? Test at a 3 percent level of significance.

Another fourth grader also grew bean plants, but in a jelly-like mass. The heights were (in inches) 24, 28, 25, 30, and 32. Do a one-way ANOVA test on the four groups. Are the heights of the bean plants different? Use the same method as shown in Example 13.4.

### Collaborative Exercise

From the class, create four groups of the same size as follows: men under 22, men at least 22, women under 22, women at least 22. Have each member of each group record the number of states in the United States he or she has visited. Run an ANOVA test to determine if the average number of states visited in the four groups are the same. Test at a 1 percent level of significance. Use one of the solution sheets in Appendix E.