1.9. The following data relate biomass production of soybeans to cumu- lative intercepted solar radiation over an eight-week period following emergence. Biomass production is the mean dry weight in grams of independent samples of four plants. (Data courtesy of Virginia Lesser and Dr. Mike Unsworth, North Carolina State University.) X Y Solar Radiation Plant Biomass 29.7 16.6 68.4 49.1 120.7 121.7 217.2 219.6 313.5 375.5 419.1 570.8 535.9 648.2 641.5 755.6 (a) Compute B and B for the linear regression of plant biomass on intercepted solar radiation. Write the regression equation. (b) Place 95% confidence intervals on 31 and 30. Interpret the in- tervals. (c) Test Ho₁ = 1.0 versus Ha B₁ 1.0 using a t-test with a = .1. Is your result for the t-test consistent with the confidence interval from Part (b)? Explain. (d) Use a t-test to test Ho: 30 = 0 against Ha: Bo 0. Interpret the results. Now fit a regression with 30 = 0. Give the analysis of variance for the regression through the origin and use an F-test to test Ho Bo 0. Compare the results of the t-test and the F-test. Do you adopt the model with or without the intercept? (e) Compute s²(31) for the regression equation without an inter- cept. Compare the variances of the estimates of the slopes for the two models. Which model provides the greater precision for the estimate of the slope? (f) Compute the 95% confidence interval estimates of the mean biomass production for X = 30 and X = 600 for both the intercept and the no-intercept models. Explain the differences in the intervals obtained for the two models. 1.12. Obtain the normal equations and the least squares estimates for the model Y₁ =+ẞixi + €i, where x = (XX). Compare the results to equation 1.6. (The model expressed in this form is referred to as the "centered" model; the independent variable has been shifted to have mean zero.) 1.13. Recompute the regression equation and analysis of variance for the Heagle ozone data (Table 1.1) using the centered model, Y₁ = μ + B₁xi +εi, where x = (XX). Compare the results with those in Tables 1.2 to 1.4. 1.14. Derive the normal equation for the no-intercept model, equation 1.40, and the least squares estimate of the slope, equation 1.41. 1.15. Derive the variance of B₁ and Ŷ; for the no-intercept model. 1.16. Show that (X-X)(-)=(X-X) = X(-Y). 1.19. The data in the table relate seed weight of soybeans, collected for six successive weeks following the start of the reproductive stage, to cumulative seasonal solar radiation for two levels of chronic ozone exposure. Seed weight is mean seed weight (grams per plant) from independent samples of four plants. (Data courtesy of Virginia Lesser and Dr. Mike Unsworth.) Low Ozone Radiation Seed Weight High Ozone Radiation Seed Weight 118.4 .7 109.1 1.3 215.2 2.9 199.6 4.8 283.9 5.6 264.2 6.5 387.9 8.7 358.2 9.4 451.5 12.4 413.2 12.9 515.6 17.4 452.5 12.3 (a) Determine the linear regression of seed weight on radiation sep- arately for each level of ozone. Determine the similarity of the two regressions by comparing the confidence interval estimates of the two intercepts and the two slopes and by visual inspection of plots of the data and the regressions. (b) Regardless of your conclusion in Part (a), assume that the two regressions are the same and estimate the common regression equation.