~ 5. Assume that the assumptions (9.9) for the linear model are valid. During the lectures we derive for the ML estimator of the slope the following result b(Y) = B(Y) · N(ẞ,0²/qxx). In order to calculate the variance, we need the following probability calculation result which is the content of this task. Assume that Y₁,..., Yn ~ N(0,σ²) H. Show by calculating that 2 E(YY)(YY) = n holds for each i, j = 1,...,n. (Hint: if EX the lecture notes is also useful). = + Jo² kun i = j kun i j 0, then E(X2) = varX. Page 35 of

Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.

Find the equation of the regression line for the following data set. x 1 2 3 y 0 3 4