4. (40 points) In an experiment, the amount of radon released in shower was investigated. Radon-enriched water was used, and five different orifice diameters were tested in shower heads. Four independent observations of amount of radon released were collected for each orifice diameter. The data from the experiment is shown in the following table (the response is the amount of radon). Orifice Diameter Mean St.D. 1.TO 89.25 0.60 75.00 1.82574186 0.80 70.50 1.29099445 1.00 74.00 1.63299316 1.20 88.50 3.31662479 (a) Write down the model which will be used to study the effect of orifice diameter on the amount of radon released (state clearly all constraints/assumptions). Explain all the terms in your model and estimate ALL the parameters. (b) Construct an upper confidence bound for the residual standard deviation. (c) Use the data to test if the average amount of radon is different across these diameters (with a=0.05). State clearly the hypotheses, test statistic, decision rule, and your conclusion. (d) Let ul, wa, us, us and us be the true means corresponding to the diameters 0.4, 0.6, 0.8, 1.0 andl.2 respectively. Let Li = ul + p2 - 2ps and La = us- 14. Estimate and construct a 95% confidence interval for LI and La, respectively. (@) Construct 95% simultaneous confidence intervals for LI and La. Use these intervals to test Ho: Li = 0 and La = 0 ve. Hi: otherwise with a=0.05. Make sure that you use a conservative method with as much power as possible. (f) Perform a pairwise comparison among the 3 different diameters by controlling the overall 0=0.05, and state your conclusion properly. Use the "best" method. () Estimate the non-centrality parameter ) of the non-central distribution of the usual ANOVA F-statistic by using your parameter estimates from part (2). (h) Using part (@), calculate the power of the test in part (@). (i) Using part (g), at least how many independent observations should be obtained for each orifice diameter (in a balanced design) such that the power of the test in part (c) would be more than 0.99? () The testing results from this data for a complete set of orthogonal contrasts {Cl, Ca, Cs, C4) are given below. Contrast DF Contrast S5 Mean Square F Value Pr > F C1 2.600000 2. 600000 0.49 0.4962 C2 03 0. 625000 0. 625000 0. 12 0.7315 CA 1. 289286 1. 289286 0.25 0.6230 Recover all the missing values for Co. Clearly indicate which one is which. Then, test if Co is significant