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Three similar methods of determining the biological oxygen demand of a waste stream are compared. Two technicians who are experienced in this type of work are available, but there is some indication that they obtain different results. A randomized block design is used, in which the blocking factor is the technician. Preliminary examination of residuals shows no systematic trends or other indication of difficulty. Results in parts per million are shown in Table 12.13. Table 12.13: Results of B.O.D. Study in parts per million Method 1 Method 2 Method 3 Technician 1 827 819 847 Technician 2 835 845 867 Is there evidence at the 5% level of significance that one or two methods of determination give higher results than the others?Concrete specimens are made using three different experimental additives. The purpose of the additives is to try to accelerate the gain of strength as the concrete sets. All specimens have the same mass ratio of additive to Portland cement, and the same mass ratio of aggregate to cement, but three different mass ratios of water to cement. Two replicate specimens are made for each of nine combinations of factors. All specimens are kept under standard conditions. After twenty-eight days the compression strengths of the specimens are measured. The results (in MPa) are shown in Table 12.9. Table 12.9: Strengths of Concrete Specimens Additives Ratio, water #1 #2 #3 to cement Compressive Strengths 0.45 40.7 42.5 40.4 39.9 41.4 41.7 0.55 36 35.6 26.6 26.3 30.7 28.2 0.65 24.7 30.6 21.9 23.9 23.9 27.6 Do these data provide evidence that the additives or the water:cement ratios or interactions of the two affect the yield strength? Use the 5% level of significance.33. Rope. To tie a boat in a harbor, how many times this as the condition for the two families to be must a rope be wound around a bollard (a vertical orthogonal (Le., to intersect at right angles)? Do your rough cylindrical post fixed on the ground) so that a graphs confirm this? man holding one end of the rope can resist a force (e) Sketch families of curves of your own choice and exerted by the boat 1000 times greater than the man find their ODEs. Can every family of curves be given can exert? First guess. Experiments show that the by an ODE? change AS of the force S in a small portion of the rope is proportional to S and to the small angle Ab 35. CAS PROJECT. Graphing Solutions. A CAS can in Fig. 16. Take the proportionality constant 0.15. usually graph solutions, even if they are integrals that The result should surprise you! cannot be evaluated by the usual analytical methods of calculus. Small (a) Show this for the five initial value problems portion V/ =e 7, )(0) = 0. $1, +2 graphing all five curves of rope on the same axes. (b) Graph approximate solution curves, using the first few terms of the Maclaurin series (obtained by term- 8 + AS wise integration of that of y') and compare with the exact curves. (c) Repeat the work in (a) for another ODE and initial conditions of your own choice, leading to an integral Fig. 16. Problem 33 that cannot be evaluated as indicated 34. TEAM PROJECT. Family of Curves. A family of 36. TEAM PROJECT. Torricelli's Law. Suppose that curves can often be characterized as the general the tank in Example 7 is hemispherical, of radius R solution of y' = Ax. J). initially full of water, and has an outlet of 5 cm cross- (a) Show that for the circles with center at the origin sectional area at the bottom. (Make a sketch.) Set we get y' = -x/y. up the model for outflow. Indicate what portion of (b) Graph some of the hyperbolas xy = c. Find an your work in Example 7 you can use (so that it can ODE for them. become part of the general method independent of the shape of the tank). Find the time / to empty the tank (c) Find an ODE for the straight lines through the (a) for any R (b) for R = 1 m. Plot / as function of origin. R Find the time when h = R/2 (a) for any R (b) for (d) You will see that the product of the right sides of R= 1 m. the ODEs in (a) and (c) equals -1. Do you recognize1. A survey of people in given region showed that 25% drank regularly. The probability of death due to liver disease, given that a person drank regularly, was 6 times the probability of death due to liver disease, given that a person did not drink regularly. The probability of death due to liver disease in the region is 0.005. If a person dies due to liver disease what is the probability that he/she drank regularly? 2. In a production line ICs are packed in vials of 5 and sent for inspection. The probabilities that the number of defectives in a vial is 0,1,2,3 are 1/3, 1/4, 1/4, 1/6 respectively. Two ICs are drawn at random from a vial and found to be good. What is the probability that all ICs in this vial are good? 3. An insurance company floats an insurance policy for an eventuality taking place with probability 0.05 over the period of policy. If the sum insured is Rs. 100000 then what should be the premium so that the expected earning of the insurance company is Rs. 1000 per policy sold? 4. Suppose X is a discrete random variable with pmf given by P(X =-1) =_ 1-2 p(X = 0) =-, P(X = 1)=- 3 3 , where a is a real number. Find the range of a for which this is a valid pmf. Also determine the maximum and minimum values of Var (X). Write the cdf for a = -. Hence show that the median is not unique. 5. Let X be a continuous random variable with the pdf b x Ve-lex 21 X =0.25) P(X 1). 18. From a store containing 2 defective, 3 partially defective and 3 good computers, a random sample of 4 computers is selected. Find the expected number of defective and partially defective computers in the sample. What will be the covariance between the number of defective and number of partially defective computers? 19. Let (X, Y) have bivariate normal distribution with density function f (x, y ) =- 4xV2 Find the correlation coefficient between X and Y, P( 1 0 . Define random variables Y1, Y2 and Y3 as Y, =X, +X, +X3, Y, = = X1 + X2 Y3 X1 X, +X2 +X3 X, +X2 Find the joint and marginal densities of Y1, Y2 and Y3. Are they independent