Life Table Applications – some examplesMarriageAssuming no marriage before age 15, no death between ages 15 and 50, and the following age specific marriage probabilities, what proportion of women will remain unmarried at their 50th birthday?Age Probability of marriage15-19 0.0520-24 0.5025-29 0.4030-34 0.2035-39 0.1540-44 0.0545-49 0.031Marriage exampleAge qx lx dx Lx Tx ex15-19 0.05 100 5 107.5 453.5 4.53520-24 0.50 95 48 167.0 346.0 3.64225-29 0.40 47 19 75.5 179.0 3.80930-34 0.20 28 6 37.0 103.5 3.69635-39 0.15 22 3 26.5 66.5 3.02340-44 0.05 19 1 20.5 40.0 2.10545-49 0.03 18 1 19.5 19.5 1.08350-54 172Example – a marriage counselorDr. X is a marriage counselor. He needs to a have a clientele of at least 100 couples for his practice to be profitable. His data show that his clients either divorce or stop seeing him in the following pattern.Year Prob. of divorce or quit therapy0 0.21 0.22 0.43 0.54 0.75 1.0Q: How many new clients does he need to get every year to ensure a profitable business?3Marriage counselor - exampleYear qx lx dx Lx Tx Ex0 0.2 100 20 90.0 257.0 2.5701 0.2 80 16 72.0 167.0 2.0882 0.4 64 26 51.0 95.0 1.4843 0.5 38 19 28.5 44.0 1.1584 0.7 19 13 12.5 15.5 0.8165 1.0 6 6 3.0 3.0 0.500he needs l0*(l0/T0)=100/(100/257) = 38.91051confirm by using 39 as radixYear qx lx dx Lx Tx Ex0 0.2 39 8 35.0 99.5 2.5511 0.2 31 6 28.0 64.5 2.0812 0.4 25 10 20.0 36.5 1.4603 0.5 15 8 11.0 16.5 1.1004 0.7 7 5 4.5 5.5 0.7865 1.0 2 2 1.0 1.0 0.5004Example – social workersA state agency provides social workers to help juvenile criminals remain in school and enter stable employment. People are eligible for these services between the ages of 14 and 20 (i.e., eligibility ceases at 20th birthday). Each social worker is given 10 new cases of 14 year-olds per year. Receipt stops when the adolescent is (a) arrested, (b) moves or otherwise disappears, (c) dies, (d) is considered “rehabilitated”, or (e) reaches age 20. The probability of stopping receipt at each age follows this pattern.Age Prob. of stopping receipt of services14 0.215 0.316 0.417 0.718 0.719 1.0Yearly expenditures for each case amount to $1,000. There are 100 social workers in the state. What is the total annual budget for these services? 5Social Workers – exampleAge qx lx dx Lx Tx Ex14 0.2 10 2 9 26 2.60015 0.3 8 2 7 17 2.12516 0.4 6 2 5 10 1.66717 0.6 4 2 3 5 1.25018 0.7 2 1 1.5 2 1.00019 1.0 1 1 0.5 0.5 0.500Each social worker thus works simultaneously with 26 youths.Total cost=(26*$1,000)*100=$2.6 million6Example – inventory and profitsA grocery store gets a shipment of 100 cartons of milk every Saturday morning (milk keeps for a week and then has to be discarded). The probability of milk being sold across the week follows this pattern:Day Probability of being soldSat 0.30Sun 0.40Mon 0.15Tues 0.20Wed 0.25Thurs 0.30Fri 0.40Each carton of milk costs the grocer $1.50 and he sells it for $2.00. It costs an average of 3 cents per day to keep one carton of milk. The grocer gets no refund on cartons that remain unsold. What is profit on a week’s shipment of milk?7Milk exampleday Qx lx dx Lx Tx exSat 0.30 100 30 145.0 450.5 4.505Sun 0.40 70 28 112.0 305.5 4.364Mon 0.15 42 6 51.0 193.5 4.607Tues 0.20 36 7 46.5 142.5 3.958Wed 0.25 29 7 39.5 96.0 3.310Thurs 0.30 22 7 32.5 56.5 2.568Fri 0.40 15 6 24.0 24.0 1.600Sat 9Total sales =(100-lsat)*$2.00 $182.00Purchase price =100*$1.50 $150.00Costs =T0*$.03 $13.52Profit =Sales-costs $18.498IUD examples from reading1) Given the life table for IUD retention in the text, what is the probability thata. a woman who retained an IUD for the first six months will still have it by the end of the 20th month?b. a woman who retained an IUD up to the beginning of the 10th month will lost it after the 18th month?c. Mrs. X who has retained her IUD until now (the beginning of the 11th month) and Mrs. Y who has also retained her IUD until now (the beginning of the 13th month) will both lose their IUDs within the next six months?9IUD example continued2) A family planning cline performs 100 IUD insertions every month. The women lose their IUD following the loss pattern in the table. Calculatea. The number of women called up per month for the special terminal medical checkup.b. The number of women clients of the clinic who at any time are wearing IUDs.c. The average number of months a woman who has just had an IUD insertion is expected to retain her