Exercise 37 of Section 3.5 gave R output for a regression of y deposition over a specified time period on two complex predictors x_{1} and x_{2} defined in terms of PAH air concentrations for various species, total time, and total amount of precipitation. Use the output in that exercise to answer the following:

a. Does there appear to be a useful linear relationship between y and at least one of the predictors?

b. The estimated standard deviation of when x_{1} is 20,000 and x_{2} is .002 is s 21.7. Calculate a 95% confidence interval for the mean value of deposition under these circumstances.

c. Fitting the model with predictors x_{1} and x_{2} gave SSResid 27,454, whereas fitting with x_{1}, x_{2}, and x_{3} x_{1}x_{2} resulted in SSResid 20519. Using .01, can we conclude that the x_{1}x_{2} term adds useful information to a ‘reduced’ model containing only x_{1} and x_{2}? Note: when g 1, the resulting F test gives the same conclusion as the t-test for whether a single variable (here, x_{1}x_{2}) contributes useful information to a model.

Exercise 37

Snowpacks contain a wide spectrum of pollutants that may represent environmental hazards. The article “Atmospheric PAH Deposition: Deposition Velocities and Washout Ratios” (J. of Envir. Engr., 2002: 186–195) focused on the deposition of polyaromatic hydrocarbons. The authors proposed a multiple regression function for relating deposition over a specified time period (y, in mg/m2 ) to two rather complicated predictors _{1} (mg-sec/m3 ) and _{2} (mg/m2 ), defined in terms of PAH air concentrations for various species, total time, and total amount of precipitation. Here is data on the species fluoranthene and corresponding output fitting from the R software:

a. Interpret the value of the coefficient of multiple determination.

b. Predict the value of deposition when _{1} 20,000 and _{1} .001.

c. Since b_{2} 29,836, is it legitimate to conclude that if _{2} increases by 1 unit while the values of the other predictors remain fixed, deposition would increase by 29,836 units? Explain your reasoning.

Q668**:**

The article “Analysis of the Modeling Methodologies for Predicting the Strength of Air-Jet Spun Yarns” (Textile Res. J., 1997: 39–44) reported on a study carried out to relate yarn tenacity (y, in g/tex) to yarn count (_{1}, in tex), percentage polyester (_{2}), first nozzle pressure (_{3}, in kg/cm^{2} ), and second nozzle pressure (x4, in kg/cm^{2} ). The estimate of the constant term in the corresponding multiple regression equation was 6.121. The estimated coefficients for the four predictors were .082, .113, .256, and .219, respectively, and the coefficient of multiple determination was .946.

a. Assuming that the sample size was n 25, state and test the appropriate hypotheses to decide whether the fitted model specifies a useful linear relationship between the dependent variable and at least one of the four model predictors.

b. Again using 25, calculate the value of adjusted R^{2} .

c. Calculate a 99% confidence interval for true mean yarn tenacity when yarn count is 16.5, yarn contains 50% polyester, first nozzle pressure is 3, and second nozzle pressure is 5 if the estimated standard deviation of predicted tenacity under these circumstances is .350.