Chapter 4: Problem 28

With one method of a procedure called acceptance sampling, a sample of items is randomly selected without replacement and the entire batch is accepted if every item in the sample is found to be okay. Exercises 27 and 28 involve acceptance sampling. Something Fishy The National Oceanic and Atmospheric Administration (NOAA) inspects seafood that is to be consumed. The inspection process involves selecting seafood samples from a larger "lot." Assume a lot contains 2875 seafood containers and 288 of these containers include seafood that does not meet inspection requirements. What is the probability that 3 selected container samples all meet requirements and the entire lot is accepted based on this sample? Does this probability seem adequate?

### Short Answer

## Step by step solution

## Define the Total and Defective Containers

## Determine the Number of Acceptable Containers

## Calculate the Probability of First Acceptable Container

## Calculate the Probability of Second Acceptable Container

## Calculate the Probability of Third Acceptable Container

## Calculate the Total Probability

## Evaluate the Result

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### probability

In this acceptance sampling exercise, we calculate the probability of specific events happening step-by-step.

For example: When we calculate the probability of the first accepted container, we use: \ \(\frac{2587}{2875}\).

This is based on the fact that there are 2587 acceptable containers out of a total of 2875 containers.

As we draw more samples, the number of total and acceptable containers decreases, leading to a series of calculated probabilities.

To find the overall probability, we multiply these individual probabilities:

\ \ \(P(\text{all three accepted}) = \frac{2587}{2875} \times \frac{2586}{2874} \times \frac{2585}{2873}\).

This multiplication gives us the combined probability of all three selected containers being acceptable.

###### statistics

In this exercise, statistical methods help determine the likelihood of sampling acceptable containers.

Using data on the total containers and defective containers, we make informed calculations on the probability of passing an acceptance sampling test.

Descriptive statistics such as proportions (acceptable vs. defective containers) form the basis for our probability calculations.

Inferential statistics, however, allow us to make predictions and generalize our findings from a sample to the population. By analyzing a small sample of containers, statistical methods help us infer the overall quality of the batch.

This example highlights the application of statistics in quality control processes to ensure standards are met.

###### quality control

For instance, NOAA inspects a sample of seafood to ensure they are safe for consumption.

If all selected samples in the batch pass the inspection, the batch is accepted. Otherwise, it is rejected.

Acceptance sampling is a form of quality control where not every item is checked, but a sample is used to infer the quality of the entire batch.

This method is efficient when dealing with large quantities, saving time and resources while maintaining quality assurance.

In our example, the goal of quality control is to ensure that accepted seafood meets inspection requirements without inspecting all 2875 containers.

###### random sampling

In acceptance sampling, random sampling helps avoid bias in selecting containers for inspection.

By randomly choosing containers, we ensure that every container has the same chance of being selected, minimizing any potential bias.

For example, when choosing 3 containers from 2875, we do so randomly to ensure that we do not inadvertently select only the best or worst containers.

Random sampling is key to the reliability of the acceptance sampling process.

It helps make sure that the inspection results give a true reflection of the batch's overall quality.

###### non-replacement sampling

This is different from replacement sampling, where the item is returned to the population after selection.

In the acceptance sampling problem, we use non-replacement sampling.

Each time a container is selected for inspection, it is not returned to the batch.

This affects the probability calculations as the total number of containers decreases with each sample drawn.

For instance, after selecting one container, the new probability for the second container is based on one fewer total and acceptable containers.

The probabilities are calculated as:

\ \ \(P(\text{second acceptable}) = \frac{2586}{2874}\) after selecting the first acceptable container.

Non-replacement sampling ensures that every draw affects the next, providing a more stringent inspection criteria.