Question

A manufacturer of hand-held calculators receives large shipments of printed circuits from a supplier. It is too costly and time-consuming to inspect all incoming circuits, so when each shipment arrives, a sample is selected for inspection. Information from the sample is then used to test \(H_{0}=\pi=.05\) versus \(H_{a}: \pi>.05\), where \(\pi\) is the true proportion of defective circuits in the shipment. If the null hypothesis is not rejected, the shipment is accepted, and the circuits are used in the production of calculators. If the null hypothesis is rejected, the entire shipment is returned to the supplier because of inferior quality. (A shipment is defined to be of inferior quality if it contains more than \(5 \%\) defective circuits.) a. In this context, define Type I and Type II errors. b. From the calculator manufacturer's point of view, which type of error is considered more serious? c. From the printed circuit supplier's point of view, which type of error is considered more serious?

Short answer

a. Type I error: The shipment is wrongly identified as of inferior quality and returned. Type II error: A shipment of inferior quality is incorrectly accepted. b. From the manufacturer's perspective, Type II error is more serious since it can result in selling defective calculators. c. From the supplier's perspective, Type I error is more serious as it can cause undeserved financial and reputation loss.

Step-by-step solution

Definition of Type I and Type II errors

a. In the context of Hypothesis testing: \n\n- A Type I error occurs when the Null Hypothesis (\(H_{0}: \pi = 0.05\)) is true, but it is rejected. This means the manufacturer wrongly identifies the shipment as of inferior quality (more than 5% defective circuits) and returns it to the supplier, while in reality it is not. \n\n- A Type II error occurs when the Null Hypothesis is false, but it is not rejected. This signifies that the manufacturer accepts the shipment assuming it to be of good quality (less than or equal to 5% defective circuits), whilst actually it contains more defective circuits i.e. it should have been rejected.

Which type of error is more serious for the manufacturer

b. From the calculator manufacturer's point of view, a Type II error is more serious. This is because, in case Type II error occurs, the manufacturer would accept a shipment of inferior quality, which means more defective calculators would be made and sold to customers, potentially damaging the manufacturer's reputation and causing a financial setback.

Which type of error is more serious for the supplier

c. From the supplier's point of view, a Type I error is more serious. That is because, if a Type I error occurs, the supplier's shipment would be incorrectly labeled as inferior and sent back, even though it met the manufacturer's quality standard. This can lead to financial losses, reputational damage and potential strain in the buyer-seller relationship.

Key concepts

Type I and Type II errors

Understanding Type I and Type II errors is fundamental in hypothesis testing, especially when dealing with quality control in manufacturing processes. Imagine a school that declares passing a test as a benchmark of a student's comprehension, similar to a manufacturer's quality check on shipments.

A Type I error, also known as a false positive, is akin to a student being wrongly failed despite understanding the material. In the context of our calculator manufacturer, this would happen if a shipment with a satisfactory proportion of defects, say exactly 5%, were erroneously rejected. Think of it as a 'false alarm' — the alarm rings without there being an actual fire. This kind of error could cause unnecessary costs and strain the relationship between the manufacturer and the supplier.

Conversely, a Type II error, or a false negative, is like a student passing without grasping the subject well. In our manufacturing scenario, this error signifies accepting a shipment that has more than 5% defective circuits, believing it meets the standards. This could lead to producing calculators that are likely to malfunction, potentially harming the manufacturer's reputation and incurring costs related to warranty and returns.

Both errors can have serious consequences, but their severity depends on the stakeholders' perspective and the specific costs associated with each error type.

Null hypothesis

The null hypothesis is a key concept in statistics, representing a default position or the assumption to be tested. To understand this, think of a court trial. The null hypothesis is like assuming the defendant is innocent until proven guilty. For our calculator manufacturer, the null hypothesis (\(H_{0}\text{: } \) is the belief that the shipment contains at most the acceptable level of 5% defective circuits, and therefore, should be accepted.

In hypothesis testing, evidence from data is used to decide whether to reject the null hypothesis. If the sample from the shipment shows more defects than the threshold, the manufacturer has grounds to reject the null hypothesis and send the shipment back. In a nutshell, the null hypothesis is the claim being scrutinized, and the evidence gathered will either reinforce this claim or support the alternative hypothesis that the proportion of defects is unsatisfactory.

Proportion of defective circuits

The quality of products, particularly in manufacturing, can be quantified by the proportion of defective circuits. This is similar to a teacher assessing the number of questions a student got wrong on a test. The manufacturer sets a threshold — exemplary in our case as 5% — to ensure productivity and brand integrity.

Statistical sampling allows manufacturers to estimate the proportion of defective circuits without checking every single item. They sample a subset of the circuits and inspect them for defects. If the sample presents defects exceeding the set threshold, it suggests that the entire batch would likely surpass the acceptable proportion of defective circuits.

The proportion of defective circuits is a crucial metric for the manufacturer. Ensuring this number stays below the threshold is paramount for maintaining quality control and customer satisfaction. By establishing this proportion, they can use hypothesis testing to make informed decisions about the quality of their suppliers' shipments, while balancing risks associated with Type I and Type II errors.