Probability assignment,
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?21. Suppose that for a certain individual, calorie intake at breakfast is random variable with mean 500 and standard deviation (s.d.) 50, calorie intake at lunch is a random variable with mean 900 and s.d. 100, and calorie intake at dinner is a random variable with mean 2000 and s.d. 180. Assuming that intakes at different meals are independent of one another, what is the approximate probability that the average calorie intake per day over the next year (365 days ) is at most 3410? 22. Let X,, X2...,X,, Xx be independent N(0, 1) random variables and Y = -J X? . Find the distribution of X.. . / VY . 23. Let X,, X,..... X2, be independent N(0,1) random variables and W =->(X2, - X_ )'. What is the distribution of W? 24. Lengths of pins (in mm) produced by a machine follow a N(#, of) distribution. Find the maximum likelihood estimators of # and o based on a random sample of size 10 with observations: 7.12, 7.13, 7.01, 6.95, 6.89, 6.97, 6.99, 6.93, 7.05, 7.02. 25. Let X,, X2...., X,, be independent N(#,o') random variables. Find unbiased estimator of #3. Is it consistent? 26. Let X,, X2. ...,X, be independent Exp(/,) random variables. Find the method of moments and maximum likelihood estimators of / and o'. Compare them by evaluating their mean squared errors. 27. Breaking strength (in kg) of the front part of a new vehicle is normally distributed. In 10 trials the breaking strengths were found to be 578, 572, 570, 568, 572, 570, 570, 572, 596, 584. Find a 95% confidence interval for the mean breaking strength. Can we say that the mean breaking strength is significantly less than 570 at 1% level of significance? Further test the hypothesis that the variance is less than 8 sq kg. At 5% level of significance. 28. Carbon emissions on 8 randomly selected vehicles of brand A were recorded as 150, 250, 240, 280, 290, 210, 220, 180, whereas those of 10 randomly selected of brand B were recorded as 140, 230, 270, 190, 270, 200, 150, 200, 190, 170. Set up 90% confidence intervals for the difference in the means and ratio of variances of the two populations. Also test the hypothesis that the variances of the two populations are equal (at 10% level of significance). Based on the result of this test, conduct an appropriate test for the hypothesis that the average emission from vehicles of brand B is less than the average emission from vehicles of brand A (at 5% level of significance). 29. An experiment was conducted to compare the recovery time (in days) of patients from a serious disease using two different medications. The first medicine was given to a random sample of 15 patients and the sample mean and sample variance of the recovery time were observed to be 16 and 1.4 respectively. The second medicine was administered to a random sample of 19 patients and sample mean and sample variance of the recovery time were observed to be 20 and 2.0 respectively. Test the hypothesis that the average recovery time using the first medicine is significantly less than the one by using the second (at 5% level). Assume the two populations to be normal with equal variances. 30. In a random sample of 200 families watching television in Bombay at any given time, it was found that 45 were watching Network A. At the same time, in a random sample of 110 families watching television in New Delhi, it was found that 32 were watching Network A. Test the hypothesis that Network A is equally popular in both states (at this time) at 1% level of significance.1. CAUTION! Constant of integration. Why is it 11-17 INITIAL VALUE PROBLEMS (IVPS) important to introduce the constant of integration immediately when you integrate? Solve the IVP. Show the steps of derivation, beginning with the general solution. 2-10 GENERAL SOLUTION 11. xy' + y= 0, M(4) = 6 Find a general solution. Show the steps of derivation. Check 12. ' = 1 + 4). MI) = 0 your answer by substitution. 2. pytr=0 13. y'cosh' x = sin y. J(0) = 17 3. y' = sec* y 14. dry di = -2tr, 1(0) = ro 4. y'sin 2 ux = Tycos 2Tx 15. y' = -4x/y, y(2) = 3 5. jy + 36x = 0 16. y' = (x + y - 2), }(0) = 2 6. V = 20-12 (Set u = x + y - 2) 7. xy = y+ 2x sin2- (Set y/x = 1) 17. xy = y+ 3x cos (1/x). )(1) = 0 Set y/ x = u) 8. y' = (y + 40)2 (Set y + 4x = v) 18. Particular solution. Introduce limits of integration in 9. xy = ] + y (Sety/x = u) (3) such that y obtained from (3) satisfies the initial 10. xy' = x + y (Set y/x = u) condition }(xQ) = Yo- SEC. 13 Separable ODEs. Modeling 19 19-36 MODELING, APPLICATIONS drying, when will it be practically dry, say, when will 19. Exponential growth. If the growth rate of the number it have lost 99% of its moisture? First guess, then of bacteria at any time / is proportional to the number calculate. present at z and doubles in 1 week, how many bacteria 28. Estimation. Could you see, practically without calcu- can be expected after 2 weeks? After 4 weeks? lation, that the answer in Prob. 27 must lie between 20. Another population model. 0 and 70 min? Explain. (a) If the birth rate and death rate of the number of 29. Alibi? Jack, arrested when leaving a bar, claims that bacteria are proportional to the number of bacteria he has been inside for at least half an hour (which present, what is the population as a function of time. would provide him with an alibi). The police check (b) What is the limiting situation for increasing time? the water temperature of his car (parked near the Interpret it. entrance of the bar) at the instant of arrest and again 30 min later, obtaining the values 190'F and 110"F. 21. Radiocarbon dating. What should be the C content respectively. Do these results give Jack an alibi? (in percent of yo) of a fossilized tree that is claimed to (Solve by inspection.) be 3000 years old? (See Example 4.) 22. Linear accelerators are used in physics for 30. Rocket. A rocket is shot straight up from the earth, accelerating charged particles. Suppose that an alpha with a net acceleration (= acceleration by the rocket particle enters an accelerator and undergoes a constant engine minus gravitational pullback) of 7/m/sec acceleration that increases the speed of the particle during the initial stage of flight until the engine cut out from 10*m/sec to 10* m/secin 10-sec. Find the at 1 = 10 sec. How high will it go, air resistance acceleration a and the distance traveled during that neglected? period of 10 sec. 31. Solution curves of y = ()/x). Show that any 23. Boyle-Mariotte's law for ideal gases." Experiments nonvertical) straight line through the origin of the show for a gas at low pressure p (and constant ryplane intersects all these curves of a given ODE at temperature) the rate of change of the volume V(p) the same angle. equals - V/p. Solve the model. 32. Friction. If a body slides on a surface, it experiences 24. Mixing problem. A tank contains 400 gal of brine friction F (a force against the direction of motion). in which 100 lb of salt are dissolved. Fresh water runs Experiments show that | | = \\N| ( Coulomb's law of into the tank at a rate of 2 gal/ min. The mixture, kept kinetic friction without lubrication), where N is the practically uniform by stirring, runs out at the same normal force (force that holds the two surfaces together; rate. How much salt will there be in the tank at the see Fig. 15) and the constant of proportionality a is end of 1 hour? called the coefficient of kinetic friction. In Fig. 15 25. Newton's law of cooling. A thermometer, reading assume that the body weighs 45 nt (about 10 lb; see 5'C, is brought into a room whose temperature is 22"C. front cover for conversion). A = 0.20 (corresponding One minute later the thermometer reading is 12"C. to steel on steel), a = 30%, the slide is 10 m long, the How long does it take until the reading is practically initial velocity is zero, and air resistance is 220 , say, 21.90 ? negligible. Find the velocity of the body at the end 26 Gompertz growth in tumors. The Gompertz model of the slide. is y' = -Ayln y (A > 0), where j(d is the mass of tumor cells at time & The model agrees well with clinical observations. The declining growth rate with increasing y > 1 corresponds to the fact that cells in the interior of a tumor may die because of insufficient My Body oxygen and nutrients. Use the ODE to discuss the growth and decline of solutions (tumors) and to find N constant solutions. Then solve the ODE. 27. Dryer. If a wet sheet in a dryer loses its moisture at a rate proportional to its moisture content, and if it loses half of its moisture during the first 10 min of Fig. 15. Problem 32
