Pollution of water resources is a serious problem that can require substantial efforts and funds to rectify. In order to determine the financial resources required, an accurate assessment of the water quality, which is measured by the index of biotic integrity (IBI), is needed. Since IBI is very expensive to measure, a study was done for a collection of streams in the Ozark Highland ecoregion of Arkansas in which the IBI for each stream was measured along with land use measures that are inexpensive to obtain. The land use measures collected in the study are the area of the watershed in square kilometers, Area, and the percent of the watershed area that is forest, Forest. The objective of this study is to determine if these land use measures can predict the IBI so that the funding required for pollution clean up can be accurately estimated. The data collected from the n = 49 watersheds are provided in the file ozark. xlsx. (a) The tab labeled "Scatterplots" displays the scatterplots to show the relationship between IBI and each of the explanatory variables, as well as a scatterplot for the two explanatory variables. Describe the relationship between each of the explanatory variables and IBI. For each explanatory variable, do you think that it will be useful in predicting the IBI?
(b) Describe the relationship between the two explanatory variables.
(c) Consider the following regression model yi = β0 + β1xi1 + β2xi2 + i . where x1 is Area and x2 is Forest. Using the tab labeled "Regression Output", report the leastsquares regression equation for this model. Interpret the estimated regression coefficients in context. Compute the predicted IBI and residual for the first data point.
(d) The tab labeled "Residual plots" display the residual plots of the regression model. Do the regression assumptions of linearity and constant variance seem to hold? Briefly explain. Note: a residual plot has the residuals on the vertical axis, and the predicted response on the horizontal axis. EXCEL produces plots of the residual against each of the explanatory variables, which should not be used in multiple linear regression. For practice, you may create your own residual plot using the directions at the end of this document.
(e) What proportion of the variation in IBI is explained by this model (do not convert to percent and round to 4 decimal places)? For practice you may want to use the sum of squares to calculate this.
(f) Researchers would like to estimate the IBI for a stream that was not a part of this study. The area of the watershed is 55 square kilometers and 84% of that watershed area is forest. Calculate a 99% prediction interval for the IBI for this stream. The standard error of ˆy at these particular values is SEyˆ = 15.9. 1 (g) Ecologists have long lobbied for protection for watershed forests because they believe that it increases the water quality. Before their next discussion with policy makers, they would like to have a estimate for the change in water quality when the percentage of watershed forest increases by 1% (assuming area of the watershed remains the same) and they ask you for an estimate with 95% confidence. What values should you give them?
(h) For practice, assume you don’t have the values given in the ANOVA table. Fill in the values in the ANOVA table, using only the regression standard error, standard deviation of the response variable, the sample size, and the number of explanatory variables. Hint: use equations (1) to (3) in Unit 11.2 slides.