Question

The article "The Epiphytic Lichen Hypogymnia physodes as a Bioindicator of Atmospheric Nitrogen and Sulphur Deposition in Norway" (Environmental Monitoring and Assessment [1993]: \(27-47\) ) gives the following data (read from a graph in the paper) on \(x=\mathrm{NO}_{3}\) wet deposition (in grams per cubic meter) and \(y=\) lichen (\% dry weight): a. What is the equation of the least-squares regression line? \(\quad \hat{y}=0.3651+0.9668 \mathrm{x}\) b. Predict lichen dry weight percentage for an \(\mathrm{NO}_{3}\) depo sition of \(0.5 \mathrm{~g} / \mathrm{m}^{3}\).

Short answer

The predicted lichen dry weight percentage for an \(NO_{3}\) deposition of \(0.5 g/m^{3}\) is approximately 0.8485 or 84.85%.

Step-by-step solution

Identify the regression line equation

The equation of the least squares regression line is already given in the problem as \(\hat{y}=0.3651+0.9668 x\), where \(x\) is the \(NO_{3}\) deposition and \(\hat{y}\) is the predicted lichen dry weight percentage. In this equation, 0.3651 is the y-intercept and 0.9668 is the slope of the line.

Plug in the given \(NO_{3}\) deposition

In order to predict the lichen dry weight percentage for an \(NO_{3}\) deposition of \(0.5 g/m^{3}\), we substitute \(x = 0.5\) into the regression equation. The calculated prediction \( \hat{y} = 0.3651+0.9668*0.5\)

Calculate the predicted lichen dry weight percentage

After substitution, we calculate the resulting mathematical expression to find the predicted lichen dry weight percentage, which comes out to be \( \hat{y} = 0.3651+0.4834 = 0.8485\). This value represents the predicted lichen dry weight percentage for an \(NO_{3}\) deposition of \(0.5 g/m^{3}\).

Key concepts

Least Squares Regression Line

The least squares regression line is a foundational tool in statistical analysis, providing a way to model the relationship between two variables. When data is scattered on a graph, this line seeks to summarize the ongoing trend by minimizing the sum of the squares of the vertical distances (errors) between the observed values and the line itself.

Imagine plotting points on a graph representing various levels of atmospheric nitrogen deposition and corresponding lichen dry weight percentages. The least squares regression line will be drawn so that the total area of the squares formed from the line to these points is as small as possible. In our exercise, the given equation \(\hat{y}=0.3651+0.9668x\) represents this best-fit line, where \(\hat{y}\) predicts the lichen dry weight percentage from any given \(NO_3\) deposition level. Here, 0.3651 is where the line intersects the Y-axis (y-intercept) and indicates the expected percentage of lichen dry weight when there is no nitrogen deposition. The slope 0.9668 signifies how much the lichen percentage increases for each gram of nitrogen deposition per cubic meter.

Predictive Analysis

Predictive analysis involves using historical data and statistical algorithms to forecast future events. In our exercise, the predictive analysis is conducted by utilizing the least squares regression line to estimate unobserved lichen dry weight percentages based on specific \(NO_3\) deposition levels. The regression line equation we have describes the best-guess relationship between the \(NO_3\) deposition and lichen dry weight percentage.

Using the equation, one can input any \(NO_3\) value and predict the corresponding lichen dry weight percentage. It's like having a crystal ball that provides scientifically backed estimates, as long as the relationship between variables remains consistent. For instance, with an \(NO_3\) deposition of 0.5 g/m³, we forecast the lichen percentage to be approximately 0.8485%. Such predictions can be critical for environmental monitoring and assessing the impact of atmospheric pollution on ecosystems.

Statistical Data Interpretation

Statistical data interpretation is the process of examining and making meaningful conclusions from data. It enables us to decipher the underlying patterns and relationships between data points. In the context of our exercise, statistical interpretation would involve understanding how lichen dry weight percentage responds or relates to varying levels of \(NO_3\) wet deposition in the environment.

By analyzing the slope and intercept of our regression line, we can comprehend not only how much lichen dry weight is expected to change with an increase in \(NO_3\) deposition but also the initial status of lichen health in the absence of \(NO_3\). In real-world scenarios, this type of analysis can inform policies and practices for environmental protection. Furthermore, by exploring the strength of the line's slope or the correlation coefficient, one can investigate the strength and direction of the relationship between these variables, often key to decision-making in environmental management and conservation.