Hypothesis Test: One-Way ANOVA

Learning Outcomes

  • Conduct a one-way ANOVA and interpret the conclusion in context

Here are some 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.
This graph has an unmarked Y axis and then an X axis that ranges from 0.00 to 4.00. It has three plot lines. The plot line labelled F subscript 1, 5 starts near the top of the Y axis at the extreme left of the graph and drops quickly to near the bottom at 0.50, at which point is slowly decreases in a curved fashion to the 4.00 mark on the X axis. The plot line labelled F subscript 100, 100 remains at Y = 0 for much of its length, except for a distinct peak between 0.50 and 1.50. The peak is a smooth curve that reaches about half way up the Y axis at its peak. The plot line labeled F subscript 5, 10 increases slightly as it progresses from 0.00 to 0.50, after which it peaks and slowly decreases down the remainder of the X axis. The peak only reaches about one fifth up the height of the Y axis.

Example 1

Let’s return to the slicing tomato exercise in the last section (try it 1). The means of the tomato yields under the five mulching conditions are represented by μ1μ2μ3μ4μ5. 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%, 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 1

MRSA, or Staphylococcus aureus, can cause serious bacterial infections in hospital patients. This table shows various colony counts from different patients who may or may not have MRSA.

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:

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.

Test whether the mean number of colonies is the same or 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% significance level.

Example 2

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

Mean Grades for Four Sororities

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%, is there a difference in mean grades among the sororities?

try it 2

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

GPAs for Four Sports Teams

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%, and determine if there is a difference in GPA among the teams.

Example 3

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 this table.

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 media in which the bean plants were grown produce the same mean height? Test at a 3% level of significance.

try it 3

Another fourth-grader also grew bean plants, but this time 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 3.

Activity

From the class, create four groups of the same size as follows: identified as men under 22, identified as men at least 22, identified as women under 22, identified as 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 is the same. Test at a 1% level of significance.