STAT 496, Spring 2004
1. Below are data on distance to failure for 38 vehicle shock absorbers.
1 For eleven of the shock absorbers there is an actual distance to failure. The remaining 27 are right censored observations, that is the last time the vehicles were checked the shock absorbers had not failed. Some possible reasons for censoring are:
•
Staggered entry. Not all vehicles are purchased at the same time.
•
Loss other than failure. Some vehicle owners do not return to the dealership to get work done on their vehicles. Or the vehicle failed due to some other reason, e.g.
a crash.
Number Distance (km) Failure?
Number Distance (km) Failure?
12
13
14
9
10
11
6
7
4
5
8
1
2
3
15
16
17
18
19
6,700
6,950
7,820
8,790
9,120
9,660
9,820
11,310
11,690
11,850
11,880
12,140
12,200
12,870
13,150
13,330
13,470
14,040
14,300
Yes
No
No
No
Yes
No
No
No
No
No
No
No
Yes
No
Yes
No
No
No
Yes
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
17,520
17,540
17,890
18,450
18,960
18,980
19,410
20,100
20,100
20,150
20,320
20,900
22,700
23,490
26,510
27,410
27,490
27,890
28,100
No
No
No
Yes
Yes
No
Yes
No
No
No
No
No
No
Yes
Yes
No
Yes
No
No
(a) Construct an estimate of the survivor function and plot this. It will be helpful if you construct a table similar to the one for the censored data bearing example given in Tape 26.
(b) Estimate the chance that a shock absorber will survive for 20,000 kilometers.
Include an approximate 95% confidence interval for this estimate.
1 O’Connor, P.D.T. (1985)
(Second edition),
Wiley, New York, p. 85
1

(c) Estimate the 50 th survivor function.
percentile (median) distance to failure using the estimated
(d) Use the values of the survivor function in (a) as the (1-p) values used in probability plotting. That is:
S
T
( t ) = (1
p )
Construct probability plots for the exponential and Weibull distributions.
(e) Which distribution, exponential or Weibull, appears to fit the data the best?
Why?
(f) From the exponential plot estimate the parameter λ .
(g) Use the value of λ from (f) to compute the probability that a shock absorber will survive longer than 20,000 kilometers.
(h) According to the exponential model, what is the median distance to failure?
(i) From the Weibull plot estimate the parameter λ and β .
(j) Use the values of λ and β from (i) to compute the probability that a shock absorber will survive longer than 20,000 kilometers.
(k) According to the Weibull model, what is the median distance to failure?
(l) How much different are the estimates of the survival probability at 20,000 kilometers and the median distance to failure for the two models?
2. Use JMP Survival - Survival/Reliability to verify your estimate of the survivor function in a) and the estimate of the 50 th percentile (median) distance in c).
3. Use JMP Survival - Parametric Regression to obtain parameter estimates as well as median distances and probabilities of surviving longer than 20,000 kilometers for both the exponential and Weibull models.
2