91Ó°ÊÓ

The following data represent the number of fish species living in various Andirondack Lakes and the \(\mathrm{pH}\) of the lakes. From chemistry, we know \(\mathrm{pH}\) is a measure of the acidity or basicity of a solution. Solutions with \(\mathrm{pH}\) less than 7 are said to be acidic. As pH increases, the solution is said to be less acidic. $$\begin{array}{lc|lc}\text { pH } & \text { Species } & \text { pH } & \text { Species } \\\\\hline 4.6 & 0 & 5.8 & 8 \\\\\hline 4.7 & 0 & 6 & 3 \\\\\hline 4.8 & 0 & 6.1 & 4 \\\\\hline 5 & 0 & 6.2 & 9 \\\\\hline 5 & 2 & 6.25 & 9 \\\\\hline 5.2 & 2 & 6.3 & 2 \\\\\hline 5.2 & 1 & 6.3 & 4 \\\\\hline 5.25 & 0 & 6.3 & 9 \\\\\hline 5.3 & 1 & 6.4 & 5 \\\\\hline 5.35 & 1 & 6.7 & 6 \\\\\hline 5.5 & 5 & 6.7 & 8 \\\\\hline 5.7 & 4 & 6.7 & 8 \\\\\hline 5.75 & 3 & 6.8 & 10\end{array}$$ (a) Draw a scatter diagram of the data treating \(\mathrm{pH}\) as the explanatory variable. (b) Determine the linear correlation coefficient between \(\mathrm{pH}\) and number of fish species. (c) Does a linear relation exist between \(\mathrm{pH}\) and number of fish species? (d) Find the least-squares regression line treating \(\mathrm{pH}\) as the explanatory variable. (e) Interpret the slope. (f) Is it reasonable to interpret the intercept? Explain. (g) What proportion of the variability in number of fish species is explained by \(\mathrm{pH} ?\) (h) Is the number of fish species in the lake whose \(\mathrm{pH}\) is 5.5 above or below average? Explain. (i) In part (g), you found the proportion of variability in number of fish species that is explained by the variability in \(\mathrm{pH}\). Can you think of other variables that might also explain the variability in the number of fish species?

Step by step solution

01

Draw a Scatter Diagram

Plot the pH values on the x-axis and the number of fish species on the y-axis. Each pair (pH, species) corresponds to a point on the scatter plot.

02

Calculate the Linear Correlation Coefficient

Use the formula for the Pearson correlation coefficient:\[ r = \frac{n\sum{(xy)} - \sum{x}\sum{y}}{ \sqrt{[n\sum{x^2} - (\sum{x})^2][n\sum{y^2} - (\sum{y})^2]} } \]where:- \(x\) is the pH value,- \(y\) is the number of fish species,- \(n\) is the number of data points.Use the given data to calculate the sums and then compute \(r\).

03

Determine if a Linear Relation Exists

Check the value of the correlation coefficient \(r\). If \(|r|\) is close to 1, a linear relation exists. Typically, values above 0.7 or below -0.7 indicate a strong correlation.

04

Find the Least-Squares Regression Line

Use the formulas for the slope \( b \) and intercept \( a \) of the regression line:\[ b = \frac{n\sum{(xy)} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2} \]\[ a = \frac{\sum{y} - b\sum{x}}{n} \]Once \( a \) and \( b \) are found, the equation of the regression line is:\[ y = a + bx \]

05

Interpret the Slope

The slope \( b \) represents the change in the number of fish species for each unit increase in pH.

06

Interpret the Intercept

The intercept \( a \) represents the expected number of fish species when the pH is 0. Evaluate whether this makes sense in the context of the problem.

07

Calculate the Proportion of Variability Explained by pH

The proportion of variability explained by pH is given by the coefficient of determination \( R^2 \), which is the square of the correlation coefficient \( r \).

08

Evaluate if Number of Fish Species for pH 5.5 is Above or Below Average

Use the regression line equation to predict the number of fish species when pH is 5.5. Compare the predicted value to the actual number of species at pH 5.5 to determine if it is above or below average.

09

Consider Other Variables

List other variables such as temperature, oxygen levels, or the presence of pollutants that might also explain variability in the number of fish species.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Regression

Linear regression is a technique used to model and analyze the relationship between two variables. In this exercise, we are examining the relationship between the pH of various lakes (explanatory variable) and the number of fish species (response variable). The goal of linear regression is to find the best-fitting line that describes this relationship. This line can help us make predictions about one variable based on the value of the other. The linear regression line is represented by the equation: \[ y = a + bx \] where \(a\) is the intercept and \(b\) is the slope. The slope indicates the rate at which the response variable changes as the explanatory variable increases.

Correlation Coefficient

The correlation coefficient, often represented by \( r \), measures the strength and direction of the linear relationship between two variables. The value of \( r \) ranges from -1 to 1:

• A value close to 1 indicates a strong positive correlation.
• A value close to -1 indicates a strong negative correlation.
• A value around 0 indicates no correlation.

In this exercise, we calculate \( r \) to see how strongly the pH of a lake is related to the number of fish species in it. A high absolute value of \( r \) (typically greater than 0.7) suggests a significant linear relationship.

Data Interpretation

Data interpretation involves analyzing the derived results to make meaningful conclusions. After plotting the scatter diagram and calculating the correlation coefficient, we can interpret these results to understand the relationship between pH and fish species. For instance, a strong positive correlation would suggest that as the pH increases (making the lake less acidic), the number of fish species tends to increase. Conversely, little to no correlation would imply that other factors might be influencing the number of fish species.

Least-Squares Method

The least-squares method is a standard approach in regression analysis to find the line of best fit. It minimizes the sum of the squares of the differences between observed values and the values predicted by the line. The goal is to make the predicted values as close as possible to the actual values. The slope \( b \) and intercept \( a \) of the regression line are calculated using formulas derived from minimizing these squared differences: \[ b = \frac{n\sum{(xy)} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2} \] \[ a = \frac{\sum{y} - b\sum{x}}{n} \] Using these, we get the equation of the regression line \( y = a + bx \).

Regression Analysis

Regression analysis is the broader statistical methodology encompassing linear regression, aimed at understanding relationships between variables. Beyond just identifying correlations, it helps predict values. In the context of this problem, regression analysis can help estimate the number of fish species for different pH levels. Through this analysis, we can generate predictions and validate the captured trends using real-world observations.

Statistical Variability

Statistical variability refers to the extent to which data points in a statistical distribution or dataset differ from each other. In the given exercise, it’s crucial to understand the variability in the number of fish species at different pH levels. This variability is captured by the coefficient of determination \( R^2 \), representing the proportion of the variance in the dependent variable that is predictable from the independent variable. An \( R^2 \) value closer to 1 indicates that a greater proportion of variance is explained by the pH variability, suggesting a more reliable model for prediction.