The following data represent the number of fish species...

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?

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?

Key Concepts

Scatter Diagram

A scatter diagram (or scatter plot) is a graphical representation of the relationship between two quantitative variables, where one variable is plotted along the horizontal axis (explanatory variable) and the other along the vertical axis (response variable). It is used in exploratory data analysis to visually assess patterns, trends, or potential linear associations between the variables.

Linear Correlation Coefficient

The linear correlation coefficient, often denoted by r, quantifies the strength and direction of the linear relationship between two quantitative variables. Its value ranges from -1 to 1, where values close to 1 indicate a strong positive linear relationship, values close to -1 indicate a strong negative linear relationship, and values near 0 suggest little to no linear association.

Least-Squares Regression Line

The least-squares regression line is a straight line that best fits the data in a scatter plot by minimizing the sum of the squared vertical distances (residuals) between the observed values and the values predicted by the line. This line describes the relationship between the explanatory and response variables and is used for prediction.

Interpretation of the Slope

The slope of the regression line represents the expected change in the response variable for a one-unit increase in the explanatory variable. It quantifies how much the dependent variable is predicted to increase or decrease when the independent variable increases by one unit.

Interpretation of the Intercept

The intercept of the regression line is the predicted value of the response variable when the explanatory variable is zero. Its interpretation depends on whether having an explanatory variable value of zero is meaningful or within the range of the data; sometimes, the intercept is mainly a mathematical artifact rather than a value with practical interpretation.

Coefficient of Determination

The coefficient of determination, denoted as R², represents the proportion of the variability in the response variable that is explained by the linear relationship with the explanatory variable. It is obtained by squaring the linear correlation coefficient and provides insight into the effectiveness of the model in explaining the observed variations in the data.

Explanatory vs. Response Variables

In regression analysis, one variable is designated as the explanatory (or independent) variable and is used to predict or explain the behavior of the response (or dependent) variable. Correctly identifying these variables is crucial for building an appropriate model and interpreting the relationship between them.

Potential Additional Variables

Besides the primary explanatory variable studied in a regression model, other factors may also affect the response variable. Recognizing that variability in a response variable could be explained by various additional predictors is essential for developing a comprehensive analysis and performing multivariate analyses when necessary.

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