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13.3 Facts About the F Distribution

Statistics13.3 Facts About the F Distribution

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 dfs.
  • 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.
The curve one the left is a nonsymmetrical F distribution curve skewed to the right, more values in the right tail and the peak is closer to the left. This curve is different from the graph on the right because of the different dfs. The curve on the right shows a nonsymmetrical F distribution curve skewed to the right. This curve is different from the graph on the left because of the different dfs. Because its dfs are larger, it more closely resembles a normal distribution curve.
Figure 13.3

Example 13.2


Let’s return to the slicing tomato exercise in Try It 13.1. The means of the tomato yields under the five mulching conditions are represented by μ1, μ2, μ3, μ4, μ5. 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.

Try It 13.2

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 = 0.6Conc = 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
Table 13.6

Plot of the data for the different concentrations:

This graph is a scatterplot for the data provided. The horizontal axis is labeled 'Colony counts' and extends from 0 - 200. The vertical axis is labeled 'Tryptone concentrations' and extends from 0.6 - 1.4.
Figure 13.5

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
Table 13.7 Mean Grades for Four Sororities


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

Try It 13.3

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
Table 13.8 GPAs for four sports teams

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
Table 13.9


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.

Try It 13.4

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.

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