Name ______________________ Mathematical Modeling in Life Sciences Spring 2017 ANSC 448 / IB 487 / STAT 458 Exam 1 (100 points) "Without passion, all the skill in the world won't lift you above craft. Without skill, all the passion in the world will leave you eager but floundering. Combining the two is the essence of the creative life." Twyla Tharp Show all work and document your results. The exam is open book and open note; however, it must be completed independently and is due by 5 P.M. Friday 17 February 2017 in Room 226 ASL or in the class folder's drop box. (Data are in 448Spr17Exam1.xls) 1. (6 points) A 0.6% phenol solution was used to disinfect a 3-hour culture of Bacillus paratyphosus, the organism that causes paratyphoid fever. Triplicate subsamples of the culture were taken at various times after the addition. The mean numbers of viable organisms were 129, 112, 81, 78, 64, 38, 19, 12, and 1 at 1, 2, 4, 5, 7, 10, 15, 20, and 32 minutes, respectively. a. What proportion of the organisms died every minute? b. If the culture contained 8 105 viable bacteria/ml when the disinfectant was added, how long was it until 99.9% of the organisms were killed? 2. (6 points) A vial of sodium phosphate solution labeled with radioactive 32P (the halflife of 32P is 14.3 days) arrived in one of our labs this morning. a. How much of this solution will have to be used in an experiment this afternoon to provide an activity of 250 Ci, given that it had an activity of 500 Ci/ml when tested 10 days ago? 3. (20 points) To improve breeding decisions in producing dog guides having an appropriate size, the following growth data (median size at a given age) for a group of male German Shepherd dogs were examined: Age, d 1 2 3 4 5 6 12 21 30 42 45 56 75 Size, kg 0.45 0.45 0.50 0.59 0.59 0.63 1.00 1.50 2.54 3.54 6.12 5.99 15.94 Age, d 135 165 195 225 255 285 315 345 375 405 435 465 495 Size, kg 24.04 25.40 26.30 26.08 27.89 26.53 29.02 31.75 27.66 29.02 28.57 28.34 29.02 105 21.63 525 28.57 a. Which of the Richards' family of growth models best describes these data? b. Interpret the parameters that you estimated for the chosen model and explain what the choice of this model implies about the growth of these dogs? 4. (4 points) Examine quantitatively, and then describe in words, the relationship between the growth of male fiddler crab (Uca pugnax) claws (chela) [ordinate, Y] and that of the rest of their body [abscissa, X] given the following data: Rest of Claw, g body, g 58 5 300 78 536 196 1080 537 1449 773 2233 1380 5. (6 points) Twinning percentage in California dairy herds currently ranges from 3 to 7% of calvings. (To demonstrate the expected range in probabilities, evaluate them for twinning percentages of both 3 and 7 %.) a. Assuming an average calving interval of 12 months and an average herd life of three lactations (three calvings), how likely is it that a single \"average\" cow never twins? b. How likely is it that an \"average\" cow has more than one set of twins in her lifetime? 6. (10 points) A sampling of plots in a young forest yielded a mean density of 2.5 tree seedlings per plot with a variance of 2.7. a. How compatible are these statistics with the assumption that a random (Poisson) spatial pattern of seedlings existed in the young forest? b. What percentages of plots would be expected to contain no seedlings, exactly 1 seedling, exactly 2 seedlings, or less than 3 seedlings if a Poisson distribution was assumed? 7. (8 points) The following table appeared on page 309 of the February 2002 issue of the Journal of Dairy Science. Table 19. The cholesterol contents of various dairy products. 1 _____________________________________________________________________ Cholesterol Identity of product Fat (%) (mg/100 g) _____________________________________________________________________ Skim milk 0.25 2 Whole milk 3.34 14 Half and half 11.50 37 Light cream 19.31 66 Medium cream 25.00 88 Nonfat dry milk 0.77 20 Cottage cheese, creamed 4.51 15 Cream cheese 34.87 110 Ice cream, vanilla 11.01 44 Blue 28.74 75 Brie 27.68 100 Cheddar 33.14 105 Mozzarella, whole milk 21.60 78 Neufchatel 23.43 76 Swiss 27.45 92 Butter 81.11 219 Sherbert orange 2.00 6 ____________________________________________________________________ 1 From Jensen and Newberg (1995); USDA Nutrient Data Base (1999). In the only reference to this table in the text, the author states: \"The amount (of cholesterol) is positively correlated with the fat content of the dairy product as shown in Table 19.\" a. Evaluate the construction of this display of data using the principles outlined in the conclusion of Tufte's analysis of the decision to launch the space shuttle Challenger. b. Provide an alternative representation of Table 19 that meets Tufte's principles. c. Quantify and discuss the appropriateness of the author's text statement. 8. (8 points) Use the applet at http://cs.unm.edu/~drew/probsci/ to demonstrate the Central Limit Theorem. Start the simulations from initial (parent) distributions of uniform, binomial, and geometric (lognormal). The default number of bins (20) can be used. Samples from the initial distribution can be varied (say 1, 5, and 10) and output visualized in the lower frame after the simulation is started and followed over time (say after 100, 200, and 500 runs). [Remember to reset between runs]. a. Summarize and interpret your results. b. Explain whether or not the same result would be expected if the initial (parent) distribution had been the Cauchy. 9. (8 points) The LD50 (lethal dose 50%) is the dose that has a 50% probability of causing death; a higher LD50 means the substance is less toxic. Typical units are milligrams per kilogram, which means that the lethal dose in milligrams is divided by the animal mass in kilograms. The test seems to be in the process of being phased out, although papers continue to appear in which it is estimated (e.g., Muralidhara et al. 2001. Acute, subacute and subchronic oral toxicity studies of 1,1-dichloroethane in rats: Application to risk evaluation. Toxicological Sciences 61:135-145. The 1,1 dichloroethane is a solvent often found as a contaminant of drinking water and a pollutant at hazardous waste sites). Muralidhara et al. (2001) reported that \"the number of fatalities in groups of 8 rats at each dosage level (in grams/kilogram) were as follows: 0 rats/8 rats - 0 g/kg; 0/8 - 1; 0/8 - 2; 0/8 - 4; 4/8 - 8; 5/8 - 12; and 8/8 - 16.\" a. Assuming that a normal distribution of mortality versus the logarithm of the dose (i.e., a lognormal distribution) is appropriate, estimate the LD50. (Note - All deaths occurred within 24 h of dosing so this portion of their data addressed only acute effects of the compound.) b. Discuss your level of confidence in the result and how you might modify a subsequent experiment to better estimate this LD50. 10. (24 points) A sample of female Svalbard reindeer had 100, 100, 100, 100, 98.1, 97.1, 95.2, 92.4, 81.9, 67.6, 49.5, 28.6, 17.1, 8.6, 2.9, 0.9, and 0.0 % survival at 1, 2, 3, ..., 17 years of age. a. Plot the percentage of survivors against age. b. Determine which model in the Richards' family of growth functions best describes these data. c. Interpret the meaning of this function's parameters. d. Derive the model you chose in part b, a function of time S(t) for the number surviving (the complement of dying) organisms in this population, given your assumption about the biological process involved. e. Compare the model you chose in part b with the function S(t) = S(0)exp(-(t/c)b) [hint: try initial values of b=0.01, c=5]. f. Summarize and discuss your results