8C In-Class Activity

Quality control, which often involves checking batches (called “lots”) of products for defects, is a very important  part of production. One method of quality  control that became commonly used  during World War II is called acceptance  sampling.[1]

Credit: iStock/NanoStockk

In acceptance sampling, a random sample is drawn from each lot of a product, and the items in the sample are tested. Each item in the sample is designated as either “conforming” to a set of standards or “nonconforming.” If the number of nonconforming items is above a pre-determined threshold, then the whole lot of the product is rejected.

Many industries today use multi-stage sampling plans if it is feasible, but we will focus on single-stage plans.[2]

Question 1

One question you might ask is, “Why not just test every item and throw out the  individual nonconforming items?” Give an example of a situation where testing every  individual item in a lot of a products might not be feasible.

 

Question 2

For acceptance sampling in quality control, the sample size and the acceptable  quality level (the maximum tolerable number of nonconforming items) will depend on  the product in question. Consider the following two scenarios:

Scenario 1: Acceptance sampling in the food industry, where frozen vegetables are  checked for the presence of a particular harmful bacteria

Scenario 2: Acceptance sampling in the clothing industry, where clothing items are  checked for structural defects

  1. Which scenario would you expect to have a lower threshold for the allowable  number of nonconforming items? Explain.
  2. Because an acceptance sample is chosen randomly, it is possible that an  unusual sample will be chosen that is not typical of the entire product lot. In  this case, the inspectors could end up rejecting a lot whose items actually fall  within the acceptable quality range, or the inspectors could accept a lot  whose items do not fall within the acceptable quality range. Which of these  situations is worse for the producer, and which is worse for the consumer?

We will consider acceptance samples as binomial experiments where the number of  successes is the number of nonconforming items in the sample. Notice that an  acceptance sample is usually drawn without replacement, so the draws are not  independent. In practice, however, lots of a product are very large, and the sample size  is small enough relative to the lot size that the independence issue is not a problem.  The population of items in the lot is so large relative to the sample size that the  probability of drawing a nonconforming item is roughly the same for each item selected, even though the sample of items is drawn without replacement. Then, the selection of  items can be considered independent, and the binomial distribution can be used to  model the situation.

For this activity, we will assume that the above description is the case for us as well— that the acceptance samples we are considering are drawn from lots of products that  are sufficiently large for us to consider our selections to be independent and to assume  that the probability of drawing a nonconforming item is the same for each item selected  for the sample.

Thus, we will consider acceptance samples drawn from lots of a product as binomial  experiments, where the number of successes is the number of nonconforming items in  the sample and a lot of products has a fixed proportion of nonconforming items.

Question 3

Explain why, in this context, an acceptance sample is an example of a binomial experiment. What will [latex]n[/latex] and [latex]p[/latex] represent for this binomial experiment?

 

Question 4

Let’s consider a situation in which acceptance samples for lots of a product have a sample size of 20, and a lot is rejected if 3 or more of the items in the sample are nonconforming.

  1. This is a binomial experiment where a “success” is a nonconforming item. Let X be the number of nonconforming items in the sample. Which of the following is a mathematical expression that describes the probability that a lot is rejected?
    1. [latex]P(X < 3)[/latex]
    2. [latex]P(X \leq 3)[/latex]
    3. [latex]P(X > 3)[/latex]
    4. [latex]P(X \geq 3)[/latex]
    5. [latex]P(X=3)[/latex]
  2. What is the probability that a lot will be rejected if the actual proportion of nonconforming items produced is 5%?

    Use the Binomial Distribution tool at https://dcmathpathways.shinyapps.io/BinomialDist/ to find this probability. Go to the Find Probabilities tab and input your values for [latex]n, p, \mbox{ and } x[/latex]. Then use the drop-down menu to select which type of probability you want.

  3. What is the probability that a lot will be rejected if the actual proportion of nonconforming items in the lot is 8%? Use the Binomial Distribution tool to find this probability.

 

Question 5

Use the data analysis tool to find the following probabilities for an acceptance sample of size 20, where the actual proportion of nonconforming items in the lot is 6%. Be sure to write down both the mathematical expression you selected in the tool (the “Type of Probability”), as well as the numerical probability the tool gives you.

  1. What is the probability that at least 1 item in the sample is nonconforming?
  2. What is the probability that less than 5 items are nonconforming?
  3. What is the probability that more than 3 items are nonconforming?
  4. What is the probability that all of the items in the sample are conforming?
  5. What is the probability that 1 to 4 items are nonconforming?

 

Question 6

If the actual proportion of nonconforming items in a lot of products was 10%, how many nonconforming items would you expect to find in a sample of 20 items? (This value is actually the mean of the binomial distribution with [latex]n = 20[/latex] and [latex]p = 0.10[/latex].)

 

Question 7

Click on the Explore tab in the Binomial Distribution tool and look at the binomial distribution when [latex]n = 20[/latex]. To answer the following questions, adjust the slider to match the given value for [latex]p[/latex].

  1. [latex]p =0.1[/latex]
    What is the shape of the graph? What do you notice about where the peak appears, especially considering your answer to Question 6?
  2. [latex]p = 0.3[/latex]
    How many items would you expect to be nonconforming in a sample of 20 items? What is the shape of the graph in this case, and where is its peak?
  3. [latex]p = 0.5[/latex]
    How many items would you expect to be nonconforming in a sample of 20 items? What is the shape of the graph in this case, and where is its peak?

 

 

  1. National Institute of Standards and Technology (n.d.).  https://www.itl.nist.gov/div898/handbook/pmc/section2/pmc22.htm ↵
  2. American Society for Quality (n.d.). Attribute & variable sampling plans and inspection procedures. https://asq.org/quality-resources/sampling/attributes-variables-sampling ↵