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Chapter 8: Q7E (page 325)

Before agreeing to purchase a large order of polyethylene sheaths for a particular type of high-pressure oil filled submarine power cable, a company wants to see conclusive evidence that the true standard deviation of

sheath thickness is less than .05 mm. What hypotheses should be tested, and why? In this context, what are the type I and type II errors?

Short Answer

Expert verified

The hypotheses are \({H_0}:\sigma = 0.05\) versus \({H_a}:\sigma < 0.05\), where \(\sigma \) is the population standard deviation.

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.

The hypotheses are \({H_0}:\sigma = 0.05\) versus \({H_a}:\sigma < 0.05\), where \(\sigma \) is the population standard deviation. Polyethylene sheaths will be used unless hypothesis \({H_0}\) is rejected, therefore the burden on proof is on the data to show that the true standard deviation of sheath thickness is not less than \(0.05\)- the sheaths will be used unless \({H_a}\) is accepted.

The type I error is to conclude that \(\sigma \)is less that \(0.05\) when it is equal to \(0.05\).

The type II error is to conclude that \(\sigma \)is \(0.05\) when it is \(0.05\).

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