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 \(H_{0}: p=.01\) versus \(H_{a}: p>.01\), where \(p\) is the actual 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 \(1 \%\) 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 would happen if the shipment is rejected even though it has less than or equals to 1% of defective circuits. A Type II error would occur if a shipment with more than 1% defective circuits is accepted. The manufacturer would regard Type II errors as more serious as this scenario could lead to faulty calculators. For the supplier, a Type I error is more severe as it would result in the unnecessary rejection of a good shipment.

Step-by-step solution

Defining Type I and Type II errors

A Type I error occurs when we reject a true null hypothesis. In this context, a Type I error would occur if the manufacturer rejects a shipment that actually has less than or equals to 1% of defective circuits. A Type II error occurs when we fail to reject a false null hypothesis. Here, a Type II error would occur if the manufacturer accepts a shipment that, in fact, has more than 1% defective circuits.

Evaluating manufacturers view

The calculator manufacturer would consider a Type II error as more serious. This is because, in a Type II error, the manufacturer would accept a shipment with more than 1% of defective circuits. This could lead to producing calculators that are faulty which could tarnish the reputation of the manufacturer and may also lead to potential financial losses.

Evaluating supplier's view

From the supplier's perspective, a Type I error would be perceived as more serious. This is because, in a Type I error, the shipment is rejected even though it meets the quality standards (i.e., it has less than or equals to 1% defective circuits). This leads to the supplier losing business due to the unjust rejection of the shipment.

Key concepts

Type I Error

A Type I Error happens when you reject the null hypothesis, even though it is actually true. In simpler terms, it's like sending back a shipment of circuits thinking they are defective, when they are actually fine.
This error is also known as a "false positive." In our context of the calculator manufacturer, a Type I Error means rejecting the shipment despite it having 1% or less defectives.
Why is this important? Because if you make a Type I Error, you could be mistakenly assuming a problem exists when it doesn't. For the supplier, this means losing a business opportunity and potential revenue because their shipment met the quality standards but was mistakenly rejected.

• Misinterprets the actual failure rate
• Leads to unnecessary extra costs and delays
• Risk of damaged business relationships due to incorrect assumptions

Being aware of Type I Errors helps in setting the right level of caution needed when testing hypotheses.

Type II Error

Type II Error occurs when you fail to reject the null hypothesis even though it is false. This is the opposite of a Type I Error.
Think of it as keeping a shipment that actually has too many defective circuits, more than 1%. You assume everything is fine when it's not, leading to production problems.
For the manufacturer, a Type II Error is considered more severe because faulty calculators may be produced, impacting quality and sales.

• False assurance of quality can lead to widespread defects
• Damages reputation and customer trust
• Potential costly recalls or rework

It highlights the importance of rigorous testing and accurate interpretation to avoid costly mistakes.

Statistical Significance

Statistical significance is a measure that helps determine if a result is not just due to random chance. It decides if the null hypothesis can be rejected.
In the context of hypothesis testing for defective circuits, statistical significance clarifies whether there is enough evidence to conclude that the proportion of defects is indeed greater than 1%.
To establish statistical significance, you often use a "p-value," which is compared against a threshold (alpha level, often 0.05). When the p-value is below this level, results are considered statistically significant.

• Provides a standard for decision-making in hypothesis testing
• Helps in minimizing errors by setting a threshold for evidence
• Assists in understanding whether observed effects are genuine

Understanding statistical significance aids both the manufacturer and supplier in making informed quality control decisions.