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Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40 -amp fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40 . If the mean amperage is lower than 40 , customers will complain because the fuses require replacement too often. If the mean amperage is higher than 40 , the manufacturer might be liable for damage to an electrical system due to fuse malfunction. To verify the amperage of the fuses, a sample of fuses is to be selected and inspected. If a hypothesis test were to be performed on the resulting data, what null and alternative hypotheses would be of interest to the manufacturer? Describe type I and type II errors in the context of this problem situation.

Step by step solution

01

Null Hypothesis Definition

The null hypothesis is a statement of no effect or no difference and is often a statement to be tested. In the context of this problem, the null hypothesis (\(H_0\)) would be that the mean amperage at which the fuses burn out is equal to 40 amps. Mathematically, this can be stated as: \(H_0: \mu = 40\), where \(\mu\) represents the mean amperage of the fuses.

02

Alternative Hypothesis Definition

The alternative hypothesis is what you might conclude if you find the null hypothesis to not be true. Here, the manufacturer is concerned with mean amperage being different from 40 amps. Thus, the alternative hypothesis (\(H_1\)) would be that the mean amperage at which the fuses burn out is not equal to 40 amps. This is represented as: \(H_1: \mu eq 40\).

03

Defining Type I Error

A Type I error occurs if the null hypothesis is rejected when it is actually true. In this context, a Type I error would be concluding that the mean amperage is not 40 amps when, in fact, it actually is 40 amps. This could lead the manufacturer to unnecessary changes in production or design.

04

Defining Type II Error

A Type II error happens if the null hypothesis is not rejected when it is actually false. For this problem, a Type II error means failing to detect that the mean amperage is not 40 amps when, in fact, it differs from 40 amps. This could result in customer complaints or potential liability for damages.

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Key Concepts

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

Null Hypothesis

In hypothesis testing, the null hypothesis is a statement that assumes no effect or difference exists, serving as a baseline for testing. For the manufacturer worried about their 40-amp fuses, the null hypothesis (H_0) would claim that the mean amperage at which the fuses burn out is exactly 40 amps. This assumption is crucial because it puts forward the idea that there is no deviation in the fuse performance.
Represented mathematically, the null hypothesis is:

• H_0: \( \mu = 40 \)

where \( \mu \) refers to the mean amperage. Ensuring the mean amperage is 40 amps adheres to safety standards and customer satisfaction. Null hypotheses provide a controlled condition—anything deviating prompts further investigation and checking if deviations are significant.

Alternative Hypothesis

The alternative hypothesis challenges the assumptions made by the null hypothesis. It suggests that there is a deviation from the norm which has to be considered. For the fuse manufacturer, the alternative hypothesis (H_1) asserts that the mean amperage is not exactly 40 amps. This hypothesis is crucial because it addresses customer and safety concerns.
Expressed mathematically, the alternative hypothesis is:

• H_1: \( \mu eq 40 \)

The alternative hypothesis here considers possibilities both below and above 40 amps. A mean amperage below 40 amps may lead to frequent replacements, whereas one above 40 amps could cause system malfunctions. Testing this hypothesis is central to maintaining product quality and customer satisfaction.

Type I Error

A Type I error, often called a "false positive," occurs when the null hypothesis is incorrectly rejected. It's like sounding a false alarm when everything is actually okay. In the scenario of the 40-amp fuses, a Type I error would imply deciding that the mean amperage is not 40 amps when, in reality, it is perfect at 40 amps.
This mistake can lead to unnecessary production adjustments or halted processes to "fix" a problem that doesn’t exist. For the manufacturer, this may mean adopting costly changes or redesigns that are actually unnecessary. Understanding the implications of a Type I error helps in devising strategies to avoid wastage.

Type II Error

A Type II error happens when the null hypothesis is falsely accepted, leading to a "false negative." It's a situation where a real issue is overlooked. For the manufacturer, it means failing to notice that the mean amperage of the fuses is not 40 amps, missing a true performance issue.
This oversight might result in ongoing customer complaints or even legal issues if the fuses cause equipment failures due to incorrect amperage ratings. Avoiding Type II errors is critical for maintaining product integrity and customer trust. Manufacturers must regularly evaluate their testing methods to minimize the risk of such errors.