Question

Biologists were interested in the distribution of trees in a wooded area. They intended to use the number of trees per 100 -square meter plot as their unit of measurement. However, they were concerned that the shapes of the plots might affect the data collection. To investigate the possibility, they counted the numbers of trees in square plots, round plots, and rectangular plots. The data are shown in the following table. ${}^{18}$ What type of analysis is appropriate for these data? $$\begin{array}{|cc|}\hline & \text{ Plot shape }\\ & \text{ Square }& \text{ Round }& \text{ Rectangular }\\ & 5& 5& 10\\ & 5& 7& 2\\ & 5& 5& 3\\ & 8& 2& 12\\ & 8& 4& 9\\ & 7& 4& 5\\ & 4& 4& 3\\ & 9& 7& 6\\ & 9& 7& 5\\ & 7& 10& 3\\ & 5& 9& 8\\ & 2& 2& 9\\ & 8& 7& 3\\ \text{ Mean }& 6.3& 5.6& 6.0\\ \text{ SD }& 2.14& 2.47& 3.27\\ \hline\end{array}$$

Short answer

An Analysis of Variance (ANOVA) is the appropriate statistical analysis to determine if the different shapes of the plots have an impact on the number of trees per unit area.

Step-by-step solution

Identify the Purpose

First, establish what the biologists aim to discover: whether the shape of the plot affects the number of trees counted.

Choose the Appropriate Statistical Test

Considering the data involve three independent samples (square, round, and rectangular plots) and the dependent variable is the number of trees, an Analysis of Variance (ANOVA) would be suitable to determine if there are any statistically significant differences between the means of these groups.

Analyzing and Interpreting the Data

Use the ANOVA test to compare the means of the three groups. The biologists would compare the F-statistic from the ANOVA to a critical value from an F-distribution table at a chosen significance level. If the F-statistic is larger than the critical value, it suggests that there are significant differences among plot shapes. The biologists would then possibly follow up with post hoc tests to identify which groups differ from each other.

Key concepts

Statistical Analysis in Biology

Statistical analysis plays a crucial role in biology, helping scientists interpret data collected from the natural world. It enables biologists to make sense of complex datasets, identify patterns, and draw conclusions about biological phenomena. In the context of the exercise, biologists are using statistical methods to understand whether different plot shapes affect tree distribution. To tackle such problems, a variety of statistical tests can be employed, each tailored to specific types of data and experimental designs.The application of these tests is guided by the nature of the variables involved. Scientists often collect data on variables that can be classified as either dependent or independent. The dependent variable, in this case, is the number of trees, as it is assumed to be affected by the independent variable, which is the shape of the plot. With the correct use of statistical analysis, biologists can ascertain whether observed differences are likely due to chance or to the effect of the independent variable on the dependent one.

ANOVA Test Application

The Analysis of Variance, more commonly known as ANOVA, is a statistical method used to compare means across multiple groups to see if at least one group mean is statistically different from the others. In biological research, where variations are inherent in living organisms and their environments, ANOVA is extremely beneficial. It addresses questions such as whether different conditions (like plot shapes) have different effects on a biological response (like the number of trees).In the exercise, the application of ANOVA is evident as the biologists have three independent groups based on plot shape. They are looking to see if these groups differ significantly in terms of the number of trees. By applying ANOVA, they can determine if the variations in tree numbers are significant when accounting for the variability within and among the different plot shapes. This test is a cornerstone in the field of experimental biology, as it allows for comparisons that take into account the natural variability of biological data.

F-distribution

The F-distribution is a probability distribution that is central to the ANOVA test. It is used to determine the critical value that the F-statistic, calculated from the ANOVA, must exceed for the differences among group means to be considered statistically significant. The shape of the F-distribution is determined by degrees of freedom that are associated with the variability within groups and among groups.When conducting an ANOVA, the calculated F-statistic is compared to a value from the F-distribution table that corresponds to a pre-determined significance level, usually 0.05 for a 95% confidence level. If the calculated F-statistic is larger than the table value, the null hypothesis, which states there are no differences among group means, can be rejected. This indicates that the differences observed are unlikely to be due to random chance. In the exercise, the biologists would consult an F-distribution table after calculating their F-statistic from the ANOVA to conclude if plot shape significantly impacts tree distribution.

Dependent and Independent Variables

In scientific research, understanding the relationship between dependent and independent variables is fundamental. The dependent variable is what the researcher measures or counts - it's the outcome that is 'dependent' on other factors. In our exercise, the dependent variable is the number of trees per plot because it's the subject of the biologists' inquiry and may change when plot shapes are different.On the other hand, the independent variable is what the researcher changes or categorizes subjects by. It is considered the 'cause' or 'influence' under investigation. The exercise deals with plot shape as the independent variable since it's the factor that's being altered. The biologists are looking to see if changing this variable—the shape of the plot (square, round, or rectangular)—has any effect on the number of trees that are counted. Studies like these, which carefully consider and analyze dependent and independent variables, are crucial for drawing valid conclusions about causal relationships in biology.