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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 analogous to the population census can be used. A population survey of a specified product is taken over a comparatively short interval of time and the number of units in the population at each age is recorded. The purpose of the survey is to estimate the percentage of units surviving to various ages, the mean service life expectancy when new and the mean residual life of used 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 be used to determine the optimum age to replace units. 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 depend only on the age of individuals and not on the particular calendar year in which they were born. This assumption is generally reasonable for the items considered in this study.

23-615 O 73 13

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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) 1ists 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

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TABLE 9

LIFE TABLE FOR CURTAINS AND DRAPERIES IN THE LIVING
ROOM OF ONE OWNER

(197 households in the Wilmington urbanized area within Delaware, 1954)

↑ All items assumed to have been acquired on January 1 of the year of acquisition.
*Number of curtain and drapery panels.

•The mean service-life expectancy in 6,329/1,000+.5 or 6.8 years.

•When there is no inventory and no removal, the removal rate is indeterminate. For this model a-survival rate of 1 was assumed, which yields a maximum figure. If a survival rate of 0 had been assumed, the mcan servicelife expectancy would have been 6.1 instead of 6.8 years, a decrcase of 10 per cent. If the data had been smoothed, the estimate would fall between these extremes.

Source: Pennock, Jean L. and Jaeger, Carol M., "Estimating the Service Life of Household Goods by Actuarial Methods," JASA, Vol 52 (1957).

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also the percentages of units surviving to various

ages. For example, approximately 54% of the units will

survive 5 years. 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.

Numbers in column (8) are obtained by cumulating numbers
in column (7) starting at the bottom. Hence the number
in column (8) corresponding to age 24 years is the sum
of the numbers in column (7) corresponding to ages 24

and greater.

2.2.2 Survival Distributions

A useful mathematical approximation to the life distribution for typical consumer products can be made using the Weibull distribution

where F(t) is the percentage of items failing before lifetime t. The parameter a will be 1 for items which wear out with increasing age. If the value of a is close to 3, the life distribution can be approximated by a normal

where 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

of a

2

and is denoted by the symbol 。.

The Weibull distribution parameters can be estimated

by plotting the percentage of units failing as a function

of age on Weibull probability paper.

If n is the sample size and x is the estimated mean life, then the standard error of estimating the true population mean life is on. In practice, the estimate for o is substituted for a to estimate the standard error.

2.2.3 Application of Statistical Techniques

For purposes of illustration, the above statistical techniques were somewhat modified because of the limitations of available data. These techniques have been applied to previously presented automobile, home appliance, and construction equipment data.