91Ó°ÊÓ

The ability of ecologists to identify regions of greatest species richness could have an impact on the preservation of genetic diversity, a major objective of the World Conservation Strategy. The article "Prediction of Rarities from Habitat Variables: Coastal Plain Plants on Nova Scotian Lakeshores" (Ecology [1992]: \(1852-1859\) ) used a sample of \(n=37\) lakes to obtain the estimated regression equation $$ \begin{aligned} \hat{y}=& 3.89+.033 x_{1}+.024 x_{2}+.023 x_{3} \\ &+.008 x_{4}-.13 x_{5}-.72 x_{6} \end{aligned} $$ where \(y=\) species richness, \(x_{1}=\) watershed area, \(x_{2}=\) shore width, \(x_{3}=\) drainage \((\%), x_{4}=\) water color (total color units), \(x_{5}=\) sand \((\%)\), and \(x_{6}=\) alkalinity. The coefficient of multiple determination was reported as \(R^{2}=.83\). Use a test with significance level \(.01\) to decide whether the chosen model is useful.

Step by step solution

01

Understand the Model

First, familiarize with the provided multiple linear regression model. Here, the species richness (\(y\)) is predicted based on the variables: watershed area (\(x_1\)), shore width (\(x_2\)), drainage (\(x_3\)), water color (\(x_4\)), sand content (\(x_5\)), and alkalinity (\(x_6\)). The model is considered significantly useful if it explains a significant amount of the variance in \(y\). This is measured by \(R^2\).

02

Null and Alternative Hypothesis

The null hypothesis (\(H_0\)) is that the regression model is not useful, meaning all regression coefficients except the constant are zero. The alternative hypothesis (\(H_a\)) is that at least one regression coefficient is not zero, making the model useful. Formally, these can be written as: \(H_0\): All \(\beta_i = 0\), except perhaps the constant (\(\beta_0\)). \(H_a\): At least one \(\beta_i ≠ 0\) where \(i = 1, 2, ..., 6\).

03

Test Statistic Calculation

In multiple linear regression, the usefulness of the model is often tested using the F-statistic. However, as the F-statistic value is not provided, it cannot be calculated here. Instead, the given coefficient of multiple determination (\(R^2\)) value can be used to gauge the fit of the model

04

Decision Rule

A common practice in stats is to consider a model significantly useful if more than 70-75% of the variance is explained, which is indicated by an \(R^2\) value larger than 0.7-0.75. Hence, with an \(R^2\)value of 0.83, it can be anticipated that model is significantly useful.

05

Conclusion

As the coefficient of multiple determination (\( R^2 = 0.83\)) is high and exceeds the 0.7 - 0.75 benchmark, the model can be considered as significantly useful for predicting species richness based on the given variables. However, a proper F-test should be done to formally test these hypotheses, which unfortunately cannot be done here due to unavailability of needed data.

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