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Chapter 8: Q8E (page 325)

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.

Short Answer

Expert verified

The hypotheses of interests are \({H_0}:\mu = 40\) versus \({H_a}:\mu \ne 40\), where \(\mu \) is the true average of amperage for the type of fuse.

Step by step solution

01

Errors in Hypothesis testing.

A type I error consists of rejecting the null hypothesis H0 when it is true.

A type II error involves not rejecting H0 when it is false.

02

Step 2:Test statistic.

A test statistic is a function of the sample data used as a basis for deciding whether H0 should be rejected. The selected test statistic should discriminate effectively between the two hypotheses. That is, values of the statistic that tend to result when H0 is true should be quite different from those typically observed when H0 is not true

03

Hypothesis results.a

The hypotheses of interests are \({H_0}:\mu = 40\) versus \({H_a}:\mu \ne 40\), where \(\mu \) is the true average of amperage for the type of fuse. Either direction is not good for the manufacturer. Therefore the alternative hypothesis should be different than \(40\).

The type I error is to conclude that \(\mu \) is not equal to \(40\) when it is \(40\).

The type II error is to conclude that \(\mu \) is \(40\) when it is different than \(40\).

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