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Twenty insects were used in an experiment to examine the effect on their activity level, y, of 3 standard preparations of a chemical. The insects were randomly assigned, 4 to receive each of the preparations and & to remain untreated as controls. Their activity levels were metered from vibrations in a test chamber and were as follows: Activity levels (y) Tovis Control 43 40 65 51 33 39 54 62 387 Preparation A 73 55 61 65 254 Preparation B 84 63 51 72 270 Preparation C 46 91 84 71 292 For these data > y = 1.203 . Ly= = 77.249. (1) Conduct an analysis of variance test to establish whether the data indicate significant differences amongst the results for the four treatments. [7] (ii) (a) Complete the following table of residuals for the data and analysis in part (i) above: Control ? ? 16.6 2.6 -15.4 -94 5.6 13.6 Preparation A 9.5 -8.5 -2.5 1.5 Preparation B 16.5 ? -16.5 4.5 Preparation C ? ? -2 (b] Make a rough plot of the residuals against the treatment means. (C) State the assumptions underlying the analysis of variance test conducted in part (i). (d) Comment on how well the data conform to these assumptions in the light of the residual plot. [8] (iii) It is suggested that any differences can be explained in terms of a difference between controls on the one hand and treated groups on the other. Comment on any evidence for this and state how you would formally test for this effect (but do not carry out the test). (4] [Total 191A model used for claim amounts (X, in units of [10,000) in certain circumstances has the following probability density function. fux). and cumulative distribution function. F(x): fix) = 5(10) *20: F(x) =1- 10 (10+.x) 10+ x You are given the information that the distribution of X has mean 2.5 units ($25,000) and standard deviation 3.23 units ((32,300). (i) Describe briefly the nature of a model for claim sizes for which the standard deviation can be greater than the mean. [2] (ii] (a) Show that we can obtain a simulated observation of X by calculating * =10 (1 r)-02 -1] where / is an observation of a random variable which is uniformly distributed on (0.1). (b] Explain why we can just as well use the formula * - 10 ,702-1] to obtain a simulated observation of X. (c) Calculate the missing values for the simulated claim amounts in the table below (which has been obtained using the method in (ii)(b) above): r Claim (f) 0.7423 6,141 0.0291 10,2872 0.2770 29,272 0.5895 11,148 0.1131 54,635 0.9897 207 0.6875 7.782 0.8525 3.243 0.0016 0.5154 [5] [Total 7]