Question

A company uses the following acceptance sampling procedure. A batch is accepted if not more than two items are found to be defective in a sample of 50 items taken randomly from the batch; otherwise, the batch is rejected. If 500 batches obtained from a process that manufactures \(2 \%\) defective items are submitted to this plan, how many batches are accepted?

Short answer

The solution to how many expected batches are accepted from 500 batches can be obtained following these steps. The exact value will depend on the probabilities computed from the binomial formula.

Step-by-step solution

Define the variables

A defective item is denoted by 'success' in the binomial probability calculation. Let \(X\) be the number of 'successes' and \(n\) the number of items. Here \(n=50\) and probability of success \(p=0.02\) for each trial.

Calculate binomial probabilities

We need to find the probability of 0, 1, and 2 defective items in a sample. We use the binomial probability formula \[P(X = k) = C(n, k) * (p)^k * (1-p)^{n-k}\] where \(C(n, k)\) is the combination. Applying this formula we can calculate the probabilities \(P(X = 0)\), \(P(X = 1)\), and \(P(X = 2)\).

Find the total probability of accepting a batch

Following the rule of addition for mutually exclusive events, the total probability \(P(X \leq 2)\) of accepting a batch is found by adding the calculated probabilities: \(P(X = 0)\), \(P(X = 1)\), and \(P(X = 2)\).

Calculate number of accepted batches

Finally, to calculate the expected number of accepted batches, multiply the total probability found in Step 3 by the total number of batches, 500 in this case. We can round that number to the nearest whole number because it's not possible to accept a fraction of a batch.

Key concepts

Binomial Probability

Binomial probability is a key concept in understanding the likelihood of a specific number of "successes" or occurrences in a fixed number of independent experiments, known as trials. In the context of acceptance sampling, we consider a defective item as a "success." Imagine you are flipping a coin multiple times: binomial probability deals with scenarios like this, and calculates how likely it is to get a certain number of heads (or tails) within those flips. Binomial probability requires three main components:

• The number of trials, denoted as \(n\).
• The probability of success on any given trial, denoted as \(p\).
• The number of successes we are interested in, denoted as \(k\).

When using these components, you utilize the formula \[P(X = k) = C(n, k) \cdot p^k \cdot (1-p)^{n-k}\], where \(C(n, k)\) determines the number of ways \(k\) successes can happen in \(n\) trials.

Defective Items

Defective items, in the context of quality control and manufacturing, refer to products that do not meet specified quality standards. In our exercise, these are the 'successes' when employing binomial probability, which might initially seem confusing. However, when calculating probabilities of defectives, it aligns well owing to the logic of identifying defects in quality control.
Knowing that our defective rate is \(2\%\), we translate this into the probability of a single item being defective in a large batch. This means every item has an independent \(0.02\) probability of being defective. Recognizing and working with this small probability in each trial helps in assessing how many defective items might occur in a given sample and ultimately impacts whether a batch is accepted or rejected in acceptance sampling procedures.

Probability Calculation

The heart of solving the acceptance sampling problem is calculating the specific probabilities of finding 0, 1, or 2 defective items in a sample using the binomial formula. This cumulative probability calculation determines whether a batch is accepted or not. First, identify the probabilities for each potential outcome (0, 1, 2 defects) by plugging them into the binomial formula \[P(X = k) = C(n, k) \cdot p^k \cdot (1-p)^{n-k}\]. Calculating each of these separately ensures you understand each step. This involves:

• Calculating the probability of zero defects: \(P(X = 0)\)
• Calculating the probability of one defect: \(P(X = 1)\)
• Calculating the probability of two defects: \(P(X = 2)\)

Summing these probabilities gives the total probability of accepting the batch, because it covers all acceptable outcomes as per the acceptance criteria.

Batch Acceptance

In acceptance sampling, a batch acceptance decision determines whether a group of items from a production line is deemed suitable for shipment or sale. In our explained scenario, a batch is accepted if no more than two items are defective in a sample of 50.
This acceptance rule helps manufacturers control quality without inspecting every single item, which is highly laborious and costly. By using statistical methods like binomial probability, decisions are made more efficiently and systematically. In practice:

• Calculate the combined probability of a batch having 0, 1, or 2 defects.
• Use this probability to determine the expected number of accepted batches by multiplying it with the total number of batches considered, here 500.
• Round the result to the nearest whole number for practical batch acceptance, as fractional batches don't make sense.

This method balances between quality assurance and operational efficiency.