Chapter 10 Case Problems
10.13 Predicting water quality. Pollution of water resources is a serious problem that can require substantial efforts to improve. To determine the financial resources required, an accurate assessment of the extent of the problem is required. The index of biotic integrity (IBI) is a measure of the water quality in streams that is fairly expensive to measure. Land use measures, on the other hand, are relatively inexpensive to obtain. IBI and land use measures for a collection of streams in the Ozark Highland ecoregion of Arkansas were collected as part of a study.8 Table 10.3 gives the data for IBI and the area of the watershed in square kilometers for streams in the original sample with area less than or equal to 70 km2.
(a) Use numerical and graphical methods to describe the variable IBI. Do the same for area. Summarize your results.
(b) Plot the data and describe the relationship. Are there any outliers or unusual patterns?
(c) Give the statistical model for simple linear regression for this problem.
(d) State the null and alternative hypotheses for examining the relationship between IBI and area.
(e) Run the simple linear regression and summarize the results.
(f) Obtain the residuals and plot them versus area. Is there anything unusual in the plot?
(g) Do the residuals appear to be approximately Normal? Give reasons for your answer.
(h) Do the assumptions for the analysis of these data using the model you gave in part (c) appear to be reasonable? Explain your answer.
10.27 Do larger houses have higher prices? We expect that there is a positive correlation between the sizes of houses in the same market and their selling prices.
(a) Use the data in Table 10.5 to test this hypothesis. (State hypotheses, find the sample correlation r and the t statistic based on it, and give an approximate P-value and your conclusion.)
(b) To what extent do you think that these results would apply to other cities in the United States?
10.36 Predicting water quality from the percent of forested area. Refer to Exercise 10.14 (page 590).
(a) Find a 95% confidence interval for mean response (index of biotic integrity) corresponding to 49% forest.
(b) Find a 95% prediction interval for a future response.
(c) Write a short paragraph interpreting the meaning of the intervals in terms of Ozark Highland streams.
(d) Do you think that these results can be applied to other streams in Arkansas or in other states? Explain why or why not.