- Exercise 1 In A Maternity Ward The Weight Of Newborn Babies Is Recorded Together With The Length Of The Pregnancy Of The 1 (133.2 KiB) Viewed 40 times
- Exercise 1 In A Maternity Ward The Weight Of Newborn Babies Is Recorded Together With The Length Of The Pregnancy Of The 2 (105.24 KiB) Viewed 40 times
Exercise 1 In a maternity ward the weight of newborn babies is recorded together with the length of the pregnancy of the mother. The data below record 20 births, the weight y of the baby is in kg, the length of pregnancy in weeks. Case Weeks Weight | Case Weeks Weight Nr. y Nr. . y 1 40 2.97 11 40 2.94 2 40 3.16 12 38 2.75 3 36 2.63 13 42 3.21 4 37 2.85 14 39 2.82 5 41 3.29 15 40 3.13 6 37 2.63 16 37 2.54 7 38 3.18 17 36 2.41 8 40 3.42 18 38 2.99 9 3.32 19 39 2.88 3.23 10 36 2.73 20 40 Table 1.1 You can assume the following calculations: = 38.7. y = 2.954, S. = 60.2, Syy = 1.5353, Sxy = 7.834 i) Calculate the point estimates for the model parameters a (the intercept), 6 (the slope), 02 (the error variance) for a simple linear regression model fitted to the data. ii) Using the information that the variation between groups of repeated observations is SSB = 1.153, calculate the ANOVA table for the data in Table 1.1. iii) Illustrate the meaning of each of the terms you calculated in part ii). iv) Test the hypothesis that the simple linear regression model is true. Exercise 2 A hydraulic engineer performs some measurements on 12 types of rectangular sections water channels of the same width and with different depths of water. The engineer wants to determine the dependence between the depth (2) of the channel and the flow (y) of water (that is, the amount of water that passes at a given point within a given time interval). The following Table 2.1 shows the result of the measurements: 2 DEPTH FLOW (x) (y) 0.10 0.1 0.20 0.4 0.28 0.7 0.30 0.6 0.35 0.7 0.40 1.2 1.0 0.41 0.45 1.4 3.8 0.60 0.75 0.80 6.0 8.0 1.00 13.0 Table 2.1 Results of measure of flow (y) with different values of water depth (x) in a water channel.
The engineer decides to try the following two models to interpret the data: Model M1: y=u + v.2 + € (1) (2) Model M2: y=u + vx + wx2 + € The calculation made by the engineer are shown in the following Table 2.2. # Model M1 > M1=lm(y x) > summary (M1) Call: lm(formula = y x) Coefficients: Estimate Std. Error t value Pr(>It!) (Intercept) X *** ** *** *** ** ** Residual standard error: 1.395 on 10 degrees of freedom Multiple R-squared: 0.8893, Adjusted R-squared: 0.8782 > # Model M2 > x2=x2 > M2=lm (y x + x2) > summary (M2) Call: 3 1m(formula = y X + x2) Coefficients: Estimate Std. Error t value Pr(>ltl (Intercept) 0.7800 X -6.6051 x2 *** *** *** ***** *** *** *** Residual standard error: 0.2804 on 9 degrees of freedom Multiple R-squared: 0.996, Adjusted R-squared: 0.9951 Table 2.2 Results of the calculations on models M1 and M2. You can assume the following: For model M1: i = 0.47, y = 3.075, Sxx = 0.7832, Syy = 175.8825, Sxy = 11.068. For model M2: 12 12 12 12 ΣΥα? = 23.718, Σα? = 3.434, Σα? = 2.475, Σα = 1.981. i=1 i=1 i=1 Answer these questions i) For model M2, write the expression of the sum of squares as a function of the parameters u, v, w, ii) Using the information provided, calculate w for model M2.