Question

An experiment was conducted to study growth characteristics of 8 different provenances (regions of natural occurrence) of Gmelina arborea (a tree native to southern Asia). There are three plots available for planting, so one tree of each provenance is planted in each plot. The response variable is the diameter of each tree (in centimeters) at breast height (1.4 meters above ground).

What type of design is being used in this experiment?

Perform the appropriate analysis to evaluate the differences in mean diameter at breast height of the eight provenances.

Which provenance(s), if any, has (have) largest mean diameter at breast height?

Comment on the effectiveness of the design in increasing the efficiency of the experiment.

Provenance	Plot	Diameter
1	1	30.85
1	2	38.01
1	3	35.1
2	1	30.24
2	2	28.43
2	3	35.93
3	1	30.94
3	2	31.64
3	3	34.95
4	1	29.89
4	2	29.12
4	3	36.75
5	1	21.52
5	2	24.07
5	3	20.76
6	1	25.38
6	2	32.14
6	3	32.19
7	1	22.89
7	2	19.66
7	3	26.92
8	1	29.44
8	2	24.95
8	3	37.99

Answer 1

We use two way analysis of variance to evaluate the differences in mean diameter at breast height of the eight provenances.

INFERENCE:

The hypothesis is given by:

(a) The mean diameter of breast height for three different plot types are same.

   The mean diameter of breast height for at least two different plot types differs.

(b) The mean diameter of breast height for eight different provenance types are same.

   The mean diameter of breast height for at least two different provenance types differs.

Since the p value is 0.00215 (with respect to provenance type) and 0.01717 which is less than 0.05, hence we reject at 5% level and conclude that mean diameter of breast height for at least two different provenance types and two plot types differs.

Since we have asked to evaluate the differences in mean diameter at breast height of the eight provenance types, we take only provenance types into consideration for inference.

PAIRWISE COMPARISON:

We have found that there is significant difference in mean diameter of breast height between at least two provenance types. In order to find between which provenance types does the mean diameter of breast height differs, we use Tukey pairwise comparison of means.

INFERENCE:

Since the p-value is less than 0.05, there is a statistically significant difference in the mean diameter of breast height between provenance types 5 and 1; 7 and 1; 5 and 2; 5 and 3; 7 and 3; 5 and 4; 7 and 4 at 5% level of significance.

The second column gives the difference in mean diameter of breast height between each two provenance types compared where the positive sign indicates that the mean diameter of first provenance type is greater than the mean diameter of second provenance type.

From the second column, we could see that the provenance type 1 has the largest mean diameter at breast height when compared to other seven provenance types as other provenances have lesser mean diameter when compared to provenance type 1.