An office supply company ordered a lot of 400 printers. When the lot arrives the company inspector will randomly inspect 12 printers. If more than three printers in the sample are non-conforming, the lot will be rejected. If fewer than two printers are non-conforming, the lot will be accepted. Otherwise, a second sample of size 8 will be taken. Suppose the inspector finds two non-conforming printers in the first sample and two in the second sample. Also AQL and LTPD are 0.05 and 0.10 respectively. Let incoming quality be 4%.
i) What is the probability of accepting the lot at the first sample?
ii) What is the probability of accepting the lot at the second sample?
iii) Find AQL and ATI
### Given: - Lot size = 400 printers - First sample size = 12 printers - Second sample size = 8 printers - AQL (Acceptable Quality Level) = 0.05 (5%) - LTPD (Lot Tolerance Percent Defective) = 0.10 (10%) - Incoming quality = 4% non-conforming printers - Acceptance rule: - If more than 3 non-conforming printers are found in the first sample, the lot is rejected. - If fewer than 2 non-conforming printers are found in the first sample, the lot is accepted. - Otherwise, a second sample is taken. ### i) Probability of accepting the lot at the first sample The lot is accepted at the first sample if fewer than 2 non-conforming printers are found. The number of non-conforming printers follows a binomial distribution: \[ P(X = k) = \binom{12}{k} p^k (1-p)^{12-k} \] Where: - \(n = 12\) (sample ___ ________ ______ _________ __________ ________ _______ __________ _______ ________ ___.
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A manufacturer of men’s jeans purchases zippers in lots of 500. The jeans manufacturer uses single-sample acceptance sampling with a sample size of 10 to determine whether to accept the lot. The manufacturer uses c = 2 as the acceptance number. Suppose 3% nonconforming zippers are acceptable to the manufacturer and 8% nonconforming zippers are not acceptable.
Let incoming quality be 4%.
i) Construct an OC curve.
ii) Average outing quality (AOQ), if the rejected lots are screened and all defective zippers are replaced by non-defectives.
iii) Average total inspection (ATI)
State whether the following statements are True or False. Give reason in support of your answer:
(a) Statistical quality control (SQC) is a technique of process control only.
(b) Twenty pieces of different length of cloth contained 2, 4, 1, 3, 5, 4, 2, 7, 3, 5, 2, 2, 4, 5, 6, 4, 2, 1, 2, 4 defects respectively. To check the process is under control with respect to the number of defects, we should use np-chart.
(c) If the probabilities are not associated with the occurrence of different states of nature, then the situation is known as decision making under risk.
(d) In single sampling plan, if we increase acceptance number then the OC curve will be steeper.
(e) A system has four components connected in parallel configuration with reliability 0.2, 0.4, 0.5, 0.8. To improve the reliability of the system most, we have to replace the component which reliability is 0.2.
An office supply company ordered a lot of 400 printers. When the lot arrives the company inspector will randomly inspect 12 printers. If more than three printers in the sample are non-conforming, the lot will be rejected. If fewer than two printers are non-conforming, the lot will be accepted. Otherwise, a second sample of size 8 will be taken. Suppose the inspector finds two non-conforming printers in the first sample and two in the second sample. Also AQL and LTPD are 0.05 and 0.10 respectively. Let incoming quality be 4%.
i) What is the probability of accepting the lot at the first sample?
ii) What is the probability of accepting the lot at the second sample?
iii) Find AQL and ATI
The failure data for 40 electronic components is shown below:
| Operating Time (in hours) | 0-5 | 5-10 | 10-15 | 15-20 | 20-25 | 25-30 | ||||||||||||
| Number of Failures | 5 | 7 | 6 | 4 | 5 | 4 | ||||||||||||
| Operating Time (in hours) | 30-35 | 35-40 | 40-45 | 45-50 | ≥50 | |||||||||||||
| Number of Failures | 4 | 0 | 2 | 1 | 2 | |||||||||||||
A system has seven independent components and reliability block diagram of it shown blow:
Find reliability of the system.
At a call centre, callers have to wait till an operator is ready to take their call. To monitor this process, 5 calls were recorded every hour for the 8-hour working day. The data below shows the waiting time in seconds:
| Time | Sample Number | ||||||||||||
| 1 | 2 | 3 | 4 | 5 | |||||||||
| 9 a.m | 8 | 9 | 15 | 4 | 11 | ||||||||
| 10 | 7 | 10 | 7 | 6 | 8 | ||||||||
| 11 | 11 | 12 | 10 | 9 | 10 | ||||||||
| 12 | 12 | 8 | 6 | 9 | 12 | ||||||||
| 1 p.m. | 11 | 10 | 6 | 14 | 11 | ||||||||
| 2 | 7 | 7 | 10 | 4 | 11 | ||||||||
| 3 | 10 | 7 | 4 | 10 | 10 | ||||||||
| 4 | 8 | 11 | 11 | 11 | 7 | ||||||||
Twenty samples each of size 10 were inspected. The number of defectives detected in each of them is given below: 0, 1, 0, 3, 9, 2, 0, 7, 0, 1, 1, 0, 0, 3, 1, 0, 0, 2, 1, 0 Find the control limits for the number of defectives and establish quality standards for the future. Plot the graph and interpret.