# F Distribution and One-way Anova

## Facts About the F Distribution

The following are facts about the F distribution:
1. The curve is not symmetrical but skewed to the right.
2. There is a different curve for each set of dfs.
3. The F statistic is greater than or equal to zero.
4. As the degrees of freedom for the numerator and for the denominator get larger, the curve approximates the normal.
5. 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.

### References

ESPN. (2012). MLB standings  2012. Retrieved from http://espn.go.com/mlb/standings/_/year/2012.
Hand, D. J. et al. (1994). A Handbook of Small Datasets: Data for Fruitfly Fecundity. London: Chapman & Hall.
Hand, D. J. et al. (1994). A Handbook of Small Datasets. London: Chapman & Hall, p. 50.
Hand. A Handbook of Small Datasets. p. 118.
Mackowiak, P. A.,Wasserman, S. S., & Levine, M. M. (1992). A critical appraisal of 98.6 degrees F, the upper limit of the normal body temperature, and other legacies of Carl Reinhold August Wunderlich. Journal of the American Medical Association, 268, 15781580.
Private K12 school in San Jose, CA. (1994). Data from a fourth grade classroom.

### Chapter Review

The graph of the F distribution is always positive and skewed right, though the shape can be mounded or exponential depending on the combination of numerator and denominator degrees of freedom. The F statistic is the ratio of a measure of the variation in the group means to a similar measure of the variation within the groups. If the null hypothesis is correct, then the numerator should be small compared to the denominator. A small F statistic will result, and the area under the F curve to the right will be large, representing a large p-value. When the null hypothesis of equal group means is incorrect, then the numerator should be large compared to the denominator, giving a large F statistic and a small area (small p-value) to the right of the statistic under the F curve.
When the data have unequal group sizes (unbalanced data), then techniques from The F Distribution and the F Ratio need to be used for hand calculations. In the case of balanced data, where the groups are the same size, simplified calculations based on group means and variances may be used. In practice, software is usually employed in the analysis. As in any analysis, graphs of various sorts should be used in conjunction with numerical techniques. Always look at your data!
Use the following information to answer the next seven exercises. Four basketball teams took a random sample of players regarding how high each player can jump (in inches). The results are shown in .
Use the following information to answer the next seven exercises. A video game developer is testing a new game on three different groups. Each group represents a different target market for the game. The developer collects scores from a random sample from each group. The results are shown in .
Use the following information to answer the next three exercises. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses.

Enter the data into your calculator or computer.
State the decisions and conclusions (in complete sentences) for the following preconceived levels of α.

### Homework

Use the following information to answer the next two exercises.  lists the number of pages in four different types of magazines.