Question

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, customerswill complain because the fuses require replacement too often. If higher, the manufacturer might be liable for damage. To verify the amperage of the fuses, a sample offuses is to be selected and inspected. If a hypothesis test were to be performed on the resulting data, what null and alternative hypothesis would be of interest tothe manufacturer? Describe type I and type II errors in the context of this problem situation.

Answer 1

Concepts and reason

A hypothesis test examines two opposing hypotheses about a population: the null hypothesis and the alternative hypothesis. The null hypothesis is the statement being tested. Usually the null hypothesis is a statement of "no effect" or "no difference". The alternative hypothesis is the statement you want to be able to conclude is true.

The Type I error occurs when the null hypothesis is rejected when it is true. It is denoted by $\alpha$ .

The Type II error occurs when the null hypothesis is failed to reject when it is false. It is denoted by $\beta$ .

Fundamentals

The formula for Type I error is,

$\alpha = P\left\{ {{H_0}{\rm{ is rejected}}|{H_0}{\rm{ is true}}} \right\}$

The formula for Type II error is,

$\beta = \left\{ {{\rm{Accepting }}{H_0}|{\rm{When it is false}}} \right\}$

The error types are tabularised as follows:

Table of error types	Null Hypothesis (H0) is
TRUE	FALSE
Decision About Null Hypothesis (H0)	Reject	Type I error ‎(False Positive)	Correct Inference ‎(True Positive)
Fail to reject	Correct Inference ‎(True Negative)	Type II error ‎(False Negative)

The claim is that the mean amperage of the fuses is different from 40-amp.

The null and alternative hypotheses are,

$\begin{array}{l}\\{H_0}:\mu = 40\\\\{H_{\rm{a}}}:\mu \ne 40\\\end{array}$

Type I error is to reject null hypothesis when it is true.

A type I error would be declaring a fuse as being defective when in fact there is nothing wrong with the fuse.

Type II error is not to reject null hypothesis when it is false.

A type II error would be declaring a fuse to be satisfactory when in fact it is defective.

Ans:

The null and alternative hypotheses are,

$\begin{array}{l}\\{H_0}:\mu = 40\\\\{H_{\rm{a}}}:\mu \ne 40\\\end{array}$

A type I error would be declaring a fuse as being defective when in fact there is nothing wrong with the fuse.

A type II error would be declaring a fuse to be satisfactory when in fact it is defective.