-39 2.2.1 Estimating Service Life By Actuarial Methods Since it is impractical to record the complete life histories of a large number of units, a method analo- tion survey of a specified product is taken over a purpose of the survey is to estimate the percentage of units surviving to various ages, the mean service life units of various ages. The removal rate of units from the population can also be estimated. This information, together with acquisition cost and repair cost data can The validity of the actuarial method depends on a rather strong assumption which is that the service life of a unit is independent of the year in which the unit was manufactured. When applied to human populations, the actuarial method assumes that birth and death rates de pend only on the age of individuals and not on the par ticular calendar year in which they were born. This sidered in this study. 23-615 0.73 - 13 -40 An illustration of this actuarial method is pre sented in Table 9 and is based on a survey of curtain and drapery service life conducted in 1957. Column (1) lists the ages in years of units surveyed. Column (2) lists the number of units surviving in each age group. Column (3) lists the number of units removed during each age interval. Column (4) is the sum of units in columns (2) and (3) at each specified age and is the number of units exposed to risk during each age interval. Column (5) is the ratio of units removed to units exposed to risk in each age interval. This ratio is called the removal rate or failure rate. Column (6) lists the survival rate, which is units minus the removal rate. Column (7) is the number of units surviving to each age from a hypothetical population of 1000 units. For example, the number surviving to age 1 year is 1000 in this Table. The number surviving 'to age 2 years is 1000 times the probability of surviving to age 2 (1.e., .8652). Hence approximately 865 units will survive to age 2 years. To find the number surviving to age x + 1, take the number surviving to age x and multiply by the probability of surviving through age interval x given in column (6). Numbers in column (7) divided by 1000 are -41 TABLE 9 LITE TABLE FOR CURTAINS AND DRAPERIES IN THE LIVING ROOM OF ONE OWNER 1,000 865 6,329 5,329 4,464 8,716 3,088 2,544 2,047 1,634 1,867 1,100 833 699 565 431 297 270 243 216 189 10 11 12 13 14 15. 16 17: 18 : 19 20 21 22 . 162 135 108 81 54 27 * All Items suaumed to bene been sexquired on January 1 dl tho your al acquisition. When there is no inventory and no removal, the removal race is indocrminale. For this model prvival nie dl I was assumed, which yielda a maximum figura Il a survival rato al 0 had been numed, tbe mcan service life aspectancy would have boon 6.1 instead of 6.8 years, a decronus el 10 per cent. If the data bed boen smoothed. the climate would fall between there atromcl. Source: Pennock, Jean L. and Jaeger, Carol M., "Estimating the Service Life of Household Goods by Actuarial Methods," -42 also the percentages of units surviving to various The residual life of a unit aged x For example, a unit aged 5 years has a mean life ex pectancy of approximately 6.1 years. and greater. 2.2.2 Survival Distributions A useful mathematical approximation to the life distribu- -(at) -43 probability distribution. The mean life y in terms of the Weibull distribution parameters is H = f(1/2) 1/2 where r is the gamma function and is tabulated in statistical tables. The variance or variability of observed lifetimes is given by The standard error of observations is the square root 2 and is denoted by the symbol o. of o The Weibull distribution parameters can be estimated of age on Weibull probability paper. If n is the sample size and is the estimated mean life, ther the standard error of estimating the true population mean life is ori. In practice, the estimate for o is substituted for o to estimate the standard error. 2.2.3 Application of Statistical Techniques For purposes of illustration, the above statistical tech niques were somewhat modified because of the limitations of available data. These techniques have been applied to previously presented automobile, home appliance, and con struction equipment data. |