Question

Industrial quality control programs often include inspection of incoming materials from suppliers. If parts are purchased in large lots, a typical plan might be to select 20 parts at random from a lot and inspect them. Suppose that a lot is judged acceptable if one or fewer of these 20 parts are defective. If more than one part is defective, the lot is rejected and returned to the supplier. Find the probability of accepting lots that have each of the following (Hint: Identify success with a defective part): a. \(5 \%\) defective parts b. \(10 \%\) defective parts c. \(20 \%\) defective parts

Short answer

The calculated probabilities are the solutions to the exercise, which depend on the given percentage of defective parts. These probabilities are obtained by using the binomial distribution theorem and summing up the probabilities for one or fewer defective parts.

Step-by-step solution

Compute basic parameters

Firstly, identify basic parameters from the exercise. These include the number of trials \(n = 20\), and the acceptance criterium that one or fewer parts are defective. Notice that the probability of success \(p\) changes in each of the sub-problems (a, b, c). It's the percentage of defective parts in a lot.

Compute probabilities for sub-problem a

For sub-problem a, the defective rate is \(5 \% = 0.05\). Thus, the probability of success, which is having a defective part, is \(p = 0.05\). Use the binomial theorem to calculate the probability of accepting the lot, i.e., that the number of defective parts is one or none (zero). The binomial distribution formula is given by \[P(k; n,p) = C(n,k) \cdot (p)^k \cdot (1-p)^{n-k}\] where \(C(n, k)\) is the binomial coefficient, calculated as \[C(n,k) = n! / [k!(n-k)!]\]. Let's apply this formula for \(k = 0\) and \(k = 1\) and sum these two probabilities.

Compute probabilities for sub-problem b

Repeat Step 2 for sub-problem b, where the defective rate is \(10 \% = 0.10\), thus \(p = 0.10\). Use the binomial theorem as before to compute the sum of probabilities for \(k = 0\) and \(k = 1\).

Compute probabilities for sub-problem c

Finally, repeat Step 2 for sub-problem c, where the defective rate is \(20 \% = 0.20\), thus \(p = 0.20\). Use the binomial theorem as before to compute the sum of probabilities for \(k = 0\) and \(k = 1\).

Key concepts

Quality Control Statistics

Quality control statistics are fundamental in manufacturing and production, ensuring that products meet certain standards of quality. In the context of industrial quality control, statistical methods are used to decide whether a batch of components is satisfactory based on a sample inspection. For instance, a company might set a rule that a batch of products is acceptable if only a small proportion of sampled items are defective. This method helps in conserving resources because inspecting every single item may not be practical or cost-effective.

Statistical techniques, such as the binomial distribution, are particularly useful when dealing with yes-or-no type attributes, such as 'defective' or 'non-defective'. In practice, inspectors would randomly select a number of parts to test from a large lot. If the number of defective parts is below a predetermined threshold, the lot passes the quality check. Otherwise, it may be rejected or subjected to further inspection.

Binomial Theorem

The binomial theorem is a crucial principle in probability theory that explains the distribution of outcomes for a given number of independent experiments, or 'trials'. When the outcome of each trial has only two possible states (such as pass or fail, defective or non-defective), the probability of a specific number of successes can be calculated using the binomial distribution formula.

Incorporating the binomial theorem in quality control allows us to precisely calculate the probability of observing a certain number of defective parts in a sample. By understanding this theorem, we can find the likelihood that, given the proportion of defective items in a lot, a randomly selected sample of parts will have a satisfactory number of defects (usually set to a maximum allowable limit).

Defective Parts Inspection

Defective parts inspection is a process used to identify and segregate defective items from a production batch, ensuring that only quality products reach the end consumer. The process typically involves randomly selecting a set number of items from a lot and checking them for defects.

When implementing a sampling plan, the objective often is to balance the risk of accepting defective lots (producer's risk) against the risk of rejecting good lots (consumer's risk). In the exercise given, the lot is considered acceptable if at most one of the sampled parts is defective. This principle protects the consumer by setting strict acceptance criteria, thus ensuring high-quality products.

Binomial Distribution Formula

The binomial distribution formula provides a way to calculate the probability of obtaining a certain number of successes in a fixed number of independent trials, each with the same probability of success. The formula is expressed as:
\[P(k; n,p) = \binom{n}{k} p^k (1-p)^{n-k}\]
Here, \(P(k; n,p)\) represents the probability of getting exactly \(k\) successes in \(n\) trials, and \(p\) is the probability of success on a single trial. The term \(\binom{n}{k}\) is the binomial coefficient, representing the number of ways to choose \(k\) successes from \(n\) trials.

In the quality control scenario, this formula helps in determining the probability that a lot will be accepted or rejected, based on the defect rate and the inspection criteria. Using this formula, we can calculate the probability for a range of defective rates, assisting quality control professionals in making informed decisions about their manufacturing processes.