Question

Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired.

a. In this context, what is a Type I error?
b. In this context, what is a Type II error?
c. Which type of error would the factory owner consider more serious?
d. Which type of error might customers consider more serious?

Answer 1

a. Type I Error: Incorrectly concluding there's a significant increase in defective items when there isn't, leading to unnecessary halting of the assembly line.

b. Type II Error: Failing to detect a real increase in defective items, allowing the assembly to continue with actual defects.

c. The factory owner would consider a Type II error more serious.

d. Customers might find a Type I error more serious due to potential delays and disruptions in product availability.

In the context of the assembly line production process:

a. Type I error: Rejecting a null hypothesis (assuming there is no significant increase in defective products) when it is actually true.

This means that the production line is falsely halted due to the mistaken belief that there is a problem when there actually isn't. This can lead to unnecessary downtime, lost productivity, and increased costs.

b. Type II error: Failing to reject a null hypothesis (assuming there is no significant increase in defective products) when it is actually false.

This means that the production line continues to operate despite the presence of a problem that is causing an increase in defective products. This can lead to subpar products being shipped to customers, damaging the company's reputation and potentially leading to recalls or lawsuits.

c. The factory owner would consider a Type II error to be more serious.

A Type II error allows defective products to reach customers, which can damage the company's reputation, lead to recalls or lawsuits, and erode customer trust. While a Type I error can cause some inconvenience and expense, it is ultimately better to err on the side of caution and halt production if there is any suspicion of a problem.

d. Customers might consider a Type I error to be more serious.

Customers would prefer to receive products that are free of defects, even if it means that production is occasionally halted unnecessarily. A Type I error ensures that defective products are not shipped to customers, while a Type II error allows defective products to reach customers, which can cause inconvenience, frustration, and even safety hazards.

Answer 2

(a) Type I error in our context is that our test indicates that the proportion of defective products has increased after inspecting but in actual the proportion of defective products was small.

(b) Type II error in our context is that our test indicates that the proportion of defective products has remained small after inspecting but in actual the proportion of defective products was increased.

(c) Factory owner would consider Type 1 error more serious.

(d) Customers will consider Type II error more serious.

Explanation:

Let
`H_0` = Proportion of defective products remains small

`H_1` = Proportion of defective products increases

(a) Type I error represents that we have rejected our null hypothesis given the fact that null hypothesis is True.

Interpretation of this Type I error in our context is that our test indicates that the proportion of defective products has increased after inspecting but in actual the proportion of defective products was small.

(b) Type II error represents that we have accepted our null hypothesis given the fact that null hypothesis is False.

Interpretation of this Type II error in our context is that our test indicates that the proportion of defective products has remained small after inspecting but in actual the proportion of defective products was increased.

(c) Factory owner would consider Type 1 error more serious because after inspecting and testing he assumed that the proportion of defective products has increased due to which he will halt the assembly process till the time the problem is identified and is repaired but in actual he should continue his assembly process as in actual the proportion of defective products was small.

(d) Customers will consider Type II error more serious because after inspecting and testing factory owner assumed that the proportion of defective products is small and he will keeps on producing products and assembly process will keeps on going but in actual the proportion of defective products was increased and due to which customers will not get good quality products and they will not be able to purchase the products further.