Chapter 13.3, Problem 31E

The accompanying data on y 5 energy output (W) and x 5 temperature difference (°K) was provided by the authors of the article “Comparison of Energy and Exergy Efficiency for Solar Box and Parabolic Cookers” (J. of Energy Engr., 2007: 53–62). The article’s authors fit a cubic regression model to the data. Here is Minitab output from such a fit.

x	23.20	23.50	23.52	24.30	25.10	26.20	27.40	28.10	29.30	30.60	31.50	32.01
y	3.78	4.12	4.24	5.35	5.87	6.02	6.12	6.41	6.62	6.43	6.13	5.92
x	32.63	33.23	33.62	34.18	35.43	35.62	36.16	36.23	36.89	37.90	39.10	41.66
y	5.64	5.45	5.21	4.98	4.65	4.50	4.34	4.03	3.92	3.65	3.02	2.89

a. What proportion of observed variation in energy output can be attributed to the model relationship?

b. Fitting a quadratic model to the data results in R² = .780. Calculate adjusted R² for this model and compare to adjusted R² for the cubic model.

c. Does the cubic predictor appear to provide useful information about y over and above that provided by the linear and quadratic predictors? State and test the appropriate hypotheses.

d. When x = 30, $s_{\hat{Y}}=.0611$  Obtain a 95% CI for true average energy output in this case, and also a 95% PI for a single energy output to be observed when temperature difference is 30. [Hint: $s_{\hat{y}}=.0611$ .]

e. Interpret the hypotheses $H_0:{\mu }_{Y\cdot 35}=5$ versus $H_a:{\mu }_{Y\cdot 35}\ne 5$, and then carry out a test at significance level .05 using the fact that when x = 35, $s_{\hat{Y}}=.0523$