Question 1 It is believed that children with too little or too much melatonin are more susceptible to de- pression. There is natural variation on levels of melatonin in children and melatonin levels are thought to be related to age. In order to investigate the relationship between melatonin and age, data are available giving the age (in months) and melatonin levels in 19 healthy children. The following model was fitted the the data Data: (y, x) for i = 1,..., 19, y = Model: ya+ xi + i, i ~ melatonin, x = age N (0, 2) independent. The results of the fitted model are: Coefficients: Estimate (Intercept) 5.9053 age -0.042446 Analysis of Variance Table Df Sum Sq Regression 1 Residuals Total 21.619 17 2.308 Mean Sq F value 21.619 159.23 Pr(>F) 1.322e-09 0.136 18 23.927 Writing this model in vector matrix notation, E(Y) = X, you may assume -1 (xx) = ( 0.3552770 -0.0050221 -0.0050221 0.0000833 (a) Write down the assumptions of this normal linear model and describe how you could informally check each assumption if the data were available to you. (b) Calculate and comment on the R value. (c) Calculate the estimated standard error for the constant term a and the estimated standard error for the coefficient of age . (d) Calculate a 95% prediction interval for a future child of age 55 months and interpret this interval. 1 (e) Calculate a 95% confidence interval for the mean melatonin for a child aged 55. How does this interval compare to the interval in question d. Explain the difference between the two intervals. (f) What does the F test in the above ANOVA table tell you about the significance of the model. Use your answer to tell me if you are asked to construct a 95% confidence interval for the slope coefficient, would it include the zero or not?