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A quality-control inspector is testing a batch of printed circuit boards to see whether they are capable of performing in a high temperature environment. He knows that the boards that will survive will pass all five of the tests with probability \(98 \% .\) They will pass at least four tests with probability \(99 \%,\) and they always pass at least three. On the other hand, the boards that will not survive sometimes pass the tests as well. In fact, \(3 \%\) pass all five tests, and another \(20 \%\) pass exactly four. The rest pass at most three tests. The inspector decides that if a board passes all five tests, he will classify it as "good." Otherwise, he'll classify it as "bad." (a) What does a type I error mean in this context? (b) What is the probability of a type I error? (c) What does a type II error mean here? (d) What is the probability of a type II error?

Step by step solution

01

Understanding Type I Error

A type I error occurs when we incorrectly reject a true null hypothesis. In this context, it means the inspector classifies a 'good' board as 'bad' even though it actually survives and performs satisfactorily in high temperature environments.

02

Calculating Probability of Type I Error

For a 'good' board, a type I error happens if it does not pass all five tests. Since a 'good' board passes all five tests with a probability of 98%, the probability of a type I error is:\[1 - 0.98 = 0.02\]

03

Understanding Type II Error

A type II error occurs when we fail to reject a false null hypothesis. In this context, it means the inspector classifies a 'bad' board as 'good' even though it actually does not survive in high temperature environments.

04

Calculating Probability of Type II Error

For a 'bad' board, a type II error happens if it passes all five tests. Since 3% of 'bad' boards pass all five tests, the probability of a type II error is:\[0.03\]

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Key Concepts

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

Type I Error

In statistics, a Type I error is a false positive result. It's like saying something is present when it's not. In the scenario with the quality-control inspector, a Type I error means labeling a good printed circuit board as bad. This is an error because the board actually meets the performance requirements. Understanding this error is crucial, especially when making decisions based on test results.
The Type I error is also referred to as an "alpha error." It is critical to minimize this error because it can lead to rejecting products that are actually acceptable, resulting in unnecessary waste. In our example, the probability of this happening is quite low at 2%. This is calculated as 1 minus the probability that a good board will pass all five tests, which is 98%.

Type II Error

A Type II error occurs when a false negative result is recorded. This is akin to saying something is absent when it is, in fact, present. In the context of our problem, a Type II error occurs when the inspector classifies a bad circuit board as good. Such an error can lead to faulty products being accepted and used.
The seriousness of a Type II error lies in its potential to compromise safety and reliability. This is especially important in quality control, where ensuring the performance of products is crucial. The probability of making a Type II error in this scenario is 3%, indicating that a small portion of bad boards might slip through as good. It is calculated by considering the 3% chance that a bad board will pass all five tests.

Probability Calculation

Calculating probabilities is a fundamental part of quality control and error analysis. In this example, calculating the likelihood of Type I and Type II errors involves understanding simple probability rules. Probability is expressed as a number between 0 and 1, where 0 means an impossible event, and 1 indicates certainty.
For Type I error, you calculate the probability of a good board not passing all five tests. It's the complement of the board passing all tests, which is 98%, giving us a probability of 2% for this error (i.e., 1 - 0.98).
For Type II error, you log the probability of a bad board passing all five tests. This value is given in the exercise as 3%. By calculating these probabilities, you can better understand the performance and reliability of your quality control measures.

Quality Control

Quality control is the process of ensuring products meet specific standards and are defect-free. It involves tests and analyses to maintain the performance and quality of products during manufacturing. In our example, quality control is about ensuring printed circuit boards function in high temperature environments.
Effective quality control minimizes Type I and Type II errors. It ensures products classified as good genuinely perform well, reducing the chance of defective units being sold or good units being discarded. This balance between accuracy of classification and minimizing errors is vital.
In practice, employing thorough testing, statistical methods, and continuous monitoring helps in maintaining control over manufacturing quality, allowing for consistent product reliability and customer satisfaction.