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10 20
30 40
50 60
70 80
90 100 D
0.005 0.01
0.015 0.02
0.025 0.03
0.035 0.04
AOQ
Figure 2 . 2 4
Average outgoing quality for a double sampling plan.
Exercises 2.11
1. A day’s production of 25 television sets from a small company has 4 which have
defects and cannot be sold. The company inspects its product by selecting 2 sets; if at most 1 of these has defects, the lot is shipped. What is the probability the lot
is shipped?
2. A shipment of 1500 washers contains 400 defective and 1100 non-defective items.
Two hundred washers are chosen at random, without replacement.
a. Find the probability that exactly 90 defective items are found. b. Approximate the probability in part a by using the binomial distribution.
3. A lot of 100 fuses is inspected by a quality control engineer who tests 10 fuses
selected at random. If 2 or fewer defective fuses are discovered, the entire lot is accepted. Find the probability that the lot is accepted if it actually contains 20
defective fuses.
4. A lot of 25 items contains 4 defective items. A sample of size 2 is chosen; the lot
is accepted if the sample shows no defective items.
a. Find the probability that the lot is shipped. b. If any defective items in the sample are replaced by good items before the lot
is shipped, find the average outgoing quality.
c. Now suppose the lot contains D defective items and that the entire lot is rectified
if the sample shows any defective items. Plot the operating characteristic curve.
2.11 Acceptance Sampling Continued
155 5. A bakery has a batch of 100 cookies, 5 of which are burned. A sample of 3 cookies
is chosen and the batch put out for sale if none of the cookies in the sample is burned.
a. What is the probability that the batch of cookies is put out for sale? b. Find the average outgoing quality if any burned cookies in the sample are
replaced by good cookies.
c. Assuming that the batch contains B burned cookies and that the entire batch
is rectified if any of the cookies in the sample is burned, show the operating characteristic curve.
6. In exercise 4, suppose that the number of defective items is unknown and that a
rejected lot is subject to 100 inspection and any defective item in the population is replaced by a good item. Estimate the average outgoing quality limit from a
graph of the average outgoing quality.
7. In exercise 5, suppose that the entire batch of cookies is inspected if the sample
should reject the batch. Estimate the average outgoing quality limit from a graph.
8. In inspecting a lot of 500 items, it is desired to accept the lot if the lot contains
1 defective item with probability 0.95, and it is desired to accept the lot if the lot contains 20 defective items with probability 0.05. Suppose the lot is accepted only
if the sample contains no defective items. What sample size is necessary?
9. A producer inspects a lot of 400 items and wants the probability that the lot is
accepted if the lot contains 1 defectives to be 0.90; the consumer wants the probability a lot containing 5 defective items is accepted to be 0.60. Suppose the
lot is accepted only if the sample contains no defective items. Find the sample size so that the sampling plan meets these risks.
10. A double sampling plan is carried out from a lot of 500 items. A sample of 10 is
selected and the lot is accepted if this sample contains no unacceptable items; if this sample contains 3 or more unacceptable items, the lot is rejected. If the sample
contains 1 or 2 unacceptable items, a second sample of 20 is drawn; the lot is then accepted if the total number of unacceptable items in the 2 samples combined is at
most 3. Suppose that at any stage an unacceptable item is replaced by a good item.
a. Find the probability that a lot containing 15 unacceptable items is accepted. b. Graph the probability in part a as a function of D, the number of unacceptable
items in the lot.
c. Find the AOQL for this double sampling plan if unacceptable lots are rectified. d. Approximate the probability that the lot is accepted using the binomial
distribution.
11. A lot of 400 items containing 3 defective items is subject to the following double
sampling plan: The lot is accepted if a first sample of 5 contains no defectives; the lot is rejected if this sample contains 2 or more defectives; if the first sample contains
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