CHAPTER 7: HYPOTHESIS TESTING: DO MY SAMPLES COME FROM THE SAME POPULATION

Table of Contents

1. CHAPTER 7: HYPOTHESIS TESTING: DO MY SAMPLES COME FROM THE SAME POPULATION? PARAMETRIC DATA
   1. Exercise 1
   2. Exercise 2
   3. Exercise 3
   4. Exercise 4

Exercise 2

This question can stand alone as an exercise that examines the criteria that need to be met before using a parametric ANOVA. However, the ideas examined here are extended in interactive exercise 2 in Chapter 8.

Example W7.2: The inhibitory effect of oregano oil and selected antibiotics on certain bacteria

As part of a larger investigation an undergraduate examined the effectiveness of the essential oil from oregano in comparison with a number of antibiotics including streptomycin. Each potential bactericide was added to a small disc of paper and placed onto agar plates that had been inoculated either with Escherichia coli or Pseudomonas aeruginosa. After incubating the plates for 24 hours at 37⁰c the zone of inhibition was recorded. The zone of inhibition is the mean diameter of the area of clear agar around the discs containing the substance under test (Table W7.5). The size of the zone of inhibition indicates the area over which the bacteria have been killed or their growth inhibited. Eight replicate plates were used for each treatment.

Table W7.5: The mean diameter of the zone of inhibition (mm) produced on agar plates inoculated either with Escherichia coli or Pseudomonas aeruginosa as a result of treatment with either oregano essential oil or streptomycin

	Mean diameter of zone	of inhibition (mm)
	E. coli	P. aeruginosa
Streptomycin	17.0	12.0
	16.0	11.0
	16.0	11.0
	15.0	10.0
	16.0	10.0
	16.0	11.0
	16.0	11.0
	17.0	11.0
Oregano oil	21.0	10.0
	22.0	9.0
	21.0	10.0
	21.0	10.0
	24.0	10.0
	22.0	10.0
	20.0	10.0
	22.0	10.0

1

Q W2.1 Using the information provided in BOXES 3.1 and 3.2 decide if these data are parametric. [If you would like to save a record of your answer, please type it into this Word document instead of the text box below] As in many undergraduate studies there are insufficient replicates within any one treatment to be confident that these data are parametric. The only criterion, listed in BOX 3.2., that we can be confident is met, is criterion a. These data are measured on an interval scale (mm).

2

Q W2.2 The undergraduate decided to assume that the data were parametric and went ahead with her plans to use a parametric two-way ANOVA to analyse the data. Before she started she checked that the criteria for using this parametric test were all met. She decided they were not and so used a non parametric ANOVA instead. Was she correct? Why? [If you would like to save a record of your answer, please type it into this Word document instead of the text box below] Yes. An Fmax test indicated that the variances were not homogeneous.

The full calculation for Q W2.2 is as follows:

There are seven criteria in section 7.7.1. Two of these criteria are/may not be met.

Criteria	Met?
1. Wish to test for differences in population means	Yes.
2. Have two treatment variables.	Yes (species of bacteria and potential bactericide).
3. Have parametric data.	Possibly. We can assume this but know that this assumption is only based on the fact that the data are measured on an interval scale of measurement.
4. Allocate all items at random to each sample.	Yes. Each agar plate was allocated at random to either the streptomycin treatment or the oregano treatment.
5. Have variances that are similar (homogeneous).	This criterion is checked using an Fmax test (BOX 7.5). In this example it is clear that the variances are not homogeneous. The variances for the four samples are included in the calculation table (W7.6). Fmax calculated = 1.41071 / 0.125 = 11.28571 a (number of samples) = 4 (degrees of freedom) = 8 - 1 = 7 At p = 0.05, Fmax critical = 8.44. Therefore Fmax calculated (11.29) is greater than Fmax critical (8.44) and the variances are not homogeneous. (We show you how to calculate Fmax using statistical software in the 'full calculations - statistical software BOX 7.5 on these web pages).
6. Have the same number of replicates (observations) in each sample.	Yes. There are 8 plates for each treatment.
7. Have an orthogonal design	Yes. Both species of bacteria were tested with each of the potential bactericide treatments.

Table W7.6: Calculation of summary statistics for the mean diameter of the zone of inhibition (mm) produced on agar plates inoculated either with Escherichia coli or Pseudomonas aeruginosa as a result of treatment with either oregano essential oil or streptomycin

	Mean diameter of zone	of inhibition (mm)
	E. coli	P. aeruginosa
Streptomycin	17	12
	16	11
	16	11
	15	10
	16	10
	16	11
	16	11
	17	11
	16.125	10.875
	2083	949
	129	87
s2	0.410714285	0.410714285
n	8	8
Oil	21	10
	22	9
	21	10
	21	10
	24	10
	22	10
	20	10
	22	10
	21.625	9.875
	3751	781
	173	79
s2	1.410714286	0.125
n	8	8