Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find...
-
Upload
adrian-horton -
Category
Documents
-
view
215 -
download
0
Transcript of Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find...
![Page 1: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/1.jpg)
Stats for Engineers Lecture 11
![Page 2: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/2.jpg)
Acceptable quality level:
(consumer happy, want to accept with high probability)
Unacceptable quality level:
(consumer unhappy, want to reject with high probability)
Acceptance Sampling Summary
Producer’s Risk: reject a batch that has acceptable quality
Consumer’s Risk: accept a batch that has unacceptable quality
Operating characteristic curve
One stage plan: can use table to find number of samples and criterionTwo stage plan: more complicated, but can require fewer samples
![Page 3: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/3.jpg)
Is acceptance sampling a good way of quality testing?
It is too far downstream in the production process; better if you can identify where things are going wrong.
Problems:
It is 0/1 (i.e. defective/OK) - not efficient use of data; large samples are required.
- better to have quality measurements on a continuous scale: earlier warning of deteriorating quality and less need for large sample sizes.
Doesn’t use any information about distribution of defective rates
![Page 4: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/4.jpg)
Reliability: Exponential Distribution revision
Which of the following has an exponential distribution?
1 2 3 4 5
27%
9%
18%
45%
0%
1. The time until a new car’s engine develops a leak
2. The number of punctures in a car’s lifetime
3. The working lifetime of a new hard disk drive
4. 1 and 3 above5. None of the above
Exponential distribution gives the time until or between random independent events that happen at constant rate. (2) is a discrete distribution. (1) and (3) are times to random events, but failure rate almost certainly increases with time.
![Page 5: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/5.jpg)
Reliability
Problem: want to know the time till failure of parts
- what is the mean time till failure?
- what is the probability that an item fails before a specified time?
E.g.
If a product lasts for many years, how do you quickly get an idea of the failure time?
accelerated life testing:
Compressed-time testing: product is tested under usual conditions but more intensively than usual (e.g. a washing machine used almost continuously)
Advanced-stress testing: product is tested under harsher conditions than normal so that failure happens soon (e.g. refrigerator motor run at a higher speed than if operating within a fridge). - requires some assumptions
![Page 6: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/6.jpg)
How do you deal with items which are still working at the end of the test programme?
An example of censored data. – we don’t know all the failure times at the end of the test
Exponential data (failure rate independent of time)
- assuming a rate, can calculate probability of no failures in .
- calculate probability of getting any set of failure times (and non failures by )
- find maximum-likelihood estimator for the failure rate in terms of failure times
⇒Estimate of failurerate is �̂�=𝑛𝑓
∑𝑖
𝑡𝑖
Test components up to a time
[see notes for derivation]
For failure times , with for parts working at , and failures
![Page 7: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/7.jpg)
Example: 50 components are tested for two weeks. 20 of them fail in this time, with an average failure time of 1.2 weeks.
What is the mean time till failure assuming a constant failure rate?
Answer:
,
∑𝑖
𝑡 𝑖=20×1.2+30×2
�̂�=𝑛𝑓
∑𝑖
𝑡𝑖
⇒�̂�=𝑛 𝑓
∑𝑖
𝑡 𝑖=2084
=0.238/week
mean time till failure is estimated to be
weeks
![Page 8: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/8.jpg)
Reliability function and failure rate
Reliability function
𝑅 (𝑡 )=𝑃 (𝑇>𝑡 )=∫𝑡
∞
𝑓 (𝑥 )𝑑𝑥
For a pdf for the time till failure, define:
Probability of surviving at least till age . i.e. that failure time is later than
¿1−𝐹 (𝑡)
is the cumulative distribution function.
Failure rate
This is failure rate at time given that it survived until time : 𝜙 (𝑡 )= 𝑓 (𝑡 )𝑅 (𝑡 )
![Page 9: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/9.jpg)
Example: Find the failure rate of the Exponential distribution
Answer:
The reliability is
Failure rate, Note: is a constant
The fact that the failure rate is constant is a special “lack of ageing property” of the exponential distribution.
- But often failure rates actually increase with age.
![Page 10: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/10.jpg)
Reliability function
Which of the following could be a plot of a reliability function?( probability of surviving at least till age . i.e. that failure time is later than )
1 2 3 4
0%
25%
42%
33%
1. Wrong12. Wrong23. Correct4. wrong3
1. 2.
3. 4.𝑡𝑡
𝑡𝑡
![Page 11: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/11.jpg)
If we measure the failure rate , how do we find the pdf?
𝜙 (𝑡 )= 𝑓 (𝑡 )𝑅 (𝑡 )
=( 𝑑𝐹𝑑𝑡 )1−𝐹 (𝑡 )
⇒ ln [1−𝐹 (𝑡 ) ]=−∫0
𝑡
𝜙 (𝑡 ′ )𝑑𝑡 ′𝐹 (0 )=0
- can hence find , and hence
Example
Say failure rate measured to be a constant,
⇒ ln [1−𝐹 (𝑡 ) ]=−∫0
𝑡
𝜈 𝑑𝑡=−𝜈𝑡
⇒1−𝐹 (𝑡 )=𝑒−𝜈𝑡 ⇒ 𝑓 (𝑡 )=𝑑𝐹 (𝑡 )𝑑𝑡
=𝜈𝑒−𝜈𝑡
- Exponential distribution
⇒𝐹 (𝑡 )=1−𝑒−𝜈𝑡
¿−𝑑𝑑𝑡 [ln (1−𝐹 (𝑡 ) ) ]
![Page 12: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/12.jpg)
The Weibull distribution
- a way to model failure rates that are not constant
Failure rate:
Parameters (shape parameter) and (scale parameter)
failure rate constant, Weibull=Exponential
failure rate increases with time
failure rate decreases with time
![Page 13: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/13.jpg)
⇒ ln (1−𝐹 (𝑡 ) )=−∫0
𝑡
𝑚𝜈𝑥𝑚−1𝑑𝑥=−𝜈𝑡𝑚
Failure rate:
⇒𝐹 (𝑡 )=1−𝑒−𝜈𝑡𝑚
Reliability: Pdf:
![Page 14: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/14.jpg)
The End!
![Page 15: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/15.jpg)
![Page 16: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/16.jpg)
[Note: as from 2011 questions are out of 25 not 20]
![Page 17: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/17.jpg)
Reliability: exam question
1 2 3 4 5
14% 14%
0%
43%
29%
(
1. 1/42. 33. 44. -35. 1/3
![Page 18: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/18.jpg)
Answer
∫−∞
∞
𝑓 (𝑡 )=1⇒∫1
∞
𝑘𝑡−4 𝑑𝑡=1
∫1
∞
𝑘𝑡− 4𝑑𝑡=[ 𝑘𝑡−3−3 ]1
∞
=
⇒𝑘=3
∫𝑎
𝑏
𝑥𝑛=[ 𝑥𝑛+1
𝑛+1 ]𝑎
𝑏
![Page 19: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/19.jpg)
Answer
⟨𝑇 ⟩=∫−∞
∞
𝑡𝑓 (𝑡 )𝑑𝑡¿∫1
∞𝑘𝑡3
=[ −𝑘2𝑡 2 ]1∞
¿0−−𝑘2
=32
Know
![Page 20: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/20.jpg)
Answer
𝜙 (𝑡 )= 𝑓 (𝑡 )𝑅 (𝑡 )
𝑅 (𝑡 )=1−𝐹 (𝑡)
𝐹 (𝑡 )=∫0
𝑡
𝑓 (𝑥 ) 𝑑𝑥 ¿∫1
𝑡𝑘𝑥4𝑑𝑥¿ [ 𝑘
−3 𝑡3 ]1𝑡
¿− 𝑘3 𝑡 3
−(− 𝑘3 )=1− 1𝑡 3⇒𝑅 (𝑡 )=1−𝐹 (𝑡 )= 1
𝑡 3
⇒𝜙 (𝑡 )= 𝑓 (𝑡 )𝑅 (𝑡 )
= 𝑘𝑡 4×𝑡 3=3
𝑡(), otherwise 0
![Page 21: Stats for Engineers Lecture 11. Acceptance Sampling Summary One stage plan: can use table to find number of samples and criterion Two stage plan: more.](https://reader031.fdocuments.net/reader031/viewer/2022032703/56649cfa5503460f949cc411/html5/thumbnails/21.jpg)
Answer
𝜙 (𝑡 )