Refer to the Journal of the Institute of Engineering (Vol. 8, 2011) study of methane gas (milligrams per liter) emitted from wastewater treatment sludge, Exercises 12.42 and 12.51. Recall that two different types of treated sludge were investigated: (1) sludge without nutrients added and (2) sludge with nutrients (VHA) added. Data for n = 27 sludge specimens were collected on the variables methane gas emitted (y), treatment time (x₁), and sludge type (x₂ = if VHA added, 0 if not).

a. Write a complete 2nd-order model for E(y) as a function of x₁ and x₂.

b. Specify the null hypothesis to test in order to determine whether curvature exists in the relationship between methane gas emission and treatment time.

c. Give the equation of the reduced model to be compared to the     complete model, part a, in order to conduct the test, part b.

d. A SAS printout of the analysis is shown on the next page. Locate the p-value of the test for curvature on the printout. Interpret the results.

Data from exercise 12.42

Wastewater treatment systems are designed to maintain the chemical, physical, and biological integrity of water. These systems, however, tend to generate various greenhouse gases, such as methane (CH₄). The Journal of the Institute of Engineering (Vol. 8, 2011) published a study of the amount of methane gas (milligrams per liter) emitted from wastewater treatment sludge. Two different types of treated sludge were investigated: (1) sludge without nutrients added and (2) sludge with nutrients added. The specific nutrient studied was volatile fatty acids (VFA). Data for n = 27 sludge specimens are listed in the table. Use regression to determine if the mean amount of methane gas emitted differs for the two types of sludge, and if so, provide a 95% confidence interval for the magnitude of the difference.

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The REG Procedure Model: MODELI Dependent Vari able: CH4 Number of Observat ions Read Number of Observations Used 27 27 Ana lys is of Variance Sum of Squar es Mean Source DF Square F Value Pr > F Model Error Corrected Total 5 21 26 244222 8287.79860 252510 48844 394.65708 123.76 <.0001 Root MSE Dependent Mean Coeff Var 19.86598 140.29630 14.16002 R-Square ndj R-Sq 0.9672 0.9594 Parame ter Estimates Parameter Est imate Standard Error t Value Pr > It! Vari able DF Intercept TINE VHA TINE VHA TIMESQ TINESA VHA -95.38759 4.14559 -331.58792 21.74633 0.03639 -0.26449 76.49017 4.60805 105.95577 6.35646 0.06472 0.08974 -1.25 0.90 -3.13 3.42 0.56 -2.96 0.2261 0.3785 0.0051 0.0026 0.5799 0.0074 Test CURVE Results for Dependent Var iable CH4 Mean Square F Value Source DF Pr > F Numerator 2781.83832 394.65708 7.05 0.0045 Denom inator 21