Question 2 [50 points] Researchers are interested in identifying characteristics of the urine that may predict kidney stones. They decide to test the relationship between urine pH level and calcium concentration in millimoles per liter. Consider the following scatterplot: Urine Ph vs. Calcium Levels 15- Ph (a) Which of the following is the most plausible correlation coefficient for the relationship between urine ph and calcium concentration? (2 points) a. 0.89 -0.12 C. -0.73 d. 0.56 A simple linear regression model with urine ph as the explanatory variable and urine calcium concentration as the outcome variable is summarized below: 1m(formula = calc ~ ph, data = ur) Coefficients : Estimate Std. Error t value Pr (>|t/) (Intercept) 7. 3811 3. 0918 2. 387 0. 0194 * ph -0.5378 0. 5093 -1. 056 0. 2942 Residual standard error: 3.258 on 77 degrees of freedom Multiple R-squared: 0.01428, Adjusted R-squared: 0.001476 F-statistic: 1.115 on 1 and 77 DF, p-value: 0.2942 (b) Which of the following correctly represents the least squares regression line? (2 points) Y = 7.38 - 0.54x; b. F = 3.09 + 0.51*; c. 1 = -0.54 + 7.38x d. P = 0.51 + 3.09%;(c) Provide an interpretation of the slope. (4 points) (d) Provide an interpretation of the intercept. (4 points) (e) Identify and interpret the value of R' in this model. (4 points) (f) Use the regression equation to estimate the urine calcium concentration for an individual with a urine ph of 11. (4 points) (g) Report the test statistic and p-value for the regression slope. Is the regression slope significant at the 5% significance level? State and interpret the test conclusion. (4 points) Urine osmolality as a measure of urine concentration provides useful information on health and hydration status. The research team decides to add urine osmolality, measured in milliosmoles per kilogram of water (mOsm/kg), to the linear regression model as an additional explanatory variable. The updated model is summarized below: 1m(formula = calc - ph + osmo, data = ur) Coefficients : Estimate Std. Error t value Pr (>|tl) (Intercept) -0.462428 3. 080787 -0. 150 0. 881 ph 0. 023967 0. 455576 0. 053 0.958 osmo 0. 007246 0. 001393 5. 200 1.67e-06 # # # Residual standard error: 2.83 on 75 degrees of freedom (1 observation deleted due to missingness) Multiple R-squared: 0.2755, Adjusted R-squared: 0.2562 F-statistic: 14.26 on 2 and 75 DF, p-value: 5.64e-06 (h) Which of the following correctly represents the new least squares regression line? (2 points) a. Y = 3.08 + 0.46x1 + 0.01X2 b. F = 0.02 - 0.462x1 + 0.01x2 c. Y = -0.462 + 0.46*1 + 0.01*2 d. P = -0.462 + 0.02x, + 0.01x2