.978 0.979 0.980 0.981 0.982 0.983 0.984 0.985 0.985 Whereas in the normal distribution the mean value divides the population in half, for the exponential distribution 36.8% of the 0.986 0.987 0.988 0.988 0.989 population is above the average and 63.2% below the average. Table 0.989 0.990 0.990 0.991 0.991 1.2 shows the area under the exponential curve for different values of the ratio K = x/x, this area being shown shaded in Figure 1.9. 0.992 0.992 0.993 0.993 0.993 0.994 0.994 0.995 0.995 0.995 0.995 0.996 0.996 0.996 0.996 0.997 0.997 0.997 0.997 0.997 0.997 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.999 0.999 0.999 0.999 0.999 0.999 Event 1.2 the area beyond 0.7 is 0.497 which is the probability that the equipment will still be working after 700 weeks. 1.9.5 Weibull distribution Figure 1.9 The exponential curve This is a continuous probability distribution and is given by Equa- As an example suppose that the time between failures of a piece tion 1.43, where a is called the scale factor, B the shape factor and of equipment is found to vary exponentially. If results indicate that Y the location factor. the mean time between failures is 1000 weeks, then what is the probability that the equipment will work for 700 weeks or more (1.43) without a failure? Calculating K as 700/1000 = 0.7 then from Table Sampling 1/9 the mean . of the population can be estimated to be within Table 1.2 Area under the exponential curve from K to +00 a certain tolerance of Ms- It is given by Equation 1.44. (1.44) 1.000 0.980 0.961 0.942 0.923 0.905 0.886 0.869 0.852 0.835 0.819 0.803 0.787 0.771 0.776 Y is found from the normal curve depending on the level of 0.741 0.726 0.712 0.698 0.684 confidence we need in specifying For 1 this level is 68.26%, for y=2 it is 95.46% and for y=3 it is 99.73%. 0.670 0.657 0.644 0.631 0.619 The standard error of mean is often defined as in Equation 1.45. 0.607 0.595 0.583 0.571 0.560 0.549 0.538 0.527 0.517 0.507 (1.45) 0.497 0.487 0.477 0.468 0.458 0.449 0.440 0.432 0.423 0.415 0.407 0.399 0.391