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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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Most popular questions from this chapter

A random sample of \(150\) recent donations at a certain blood bank reveals that \(82\) were type A blood. Does this suggest that the actual percentage of type A donations differs from \(40\% \), the percentage of the population having type A blood? Carry out a test of the appropriate hypotheses using a significance level of \(.01\). Would your conclusion have been different if a significance level of \(.05\) had been used?

A manufacturer of plumbing fixtures has developed a new type of washer less faucet. Let \(p = P\) (a randomly selected faucet of this type will develop a leak within \(2\) years under normal use). The manufacturer has decided to proceed with production unless it can be determined that \(p\) is too large; the borderline acceptable value of \(p\) is specified as \(.10\). The manufacturer decides to subject \(n\) of these faucets to accelerated testing (approximating \(2\) years of normal use). With \(X = \) the number among the \(n\) faucets that leak before the test concludes, production will commence unless the observed X is too large. It is decided that if \(p = .10\), the probability of not proceeding should be at most \(.10\), whereas if \(p = .30\) the probability of proceeding should be at most \(.10\). Can \(n = 10\) be used? \(n = 20\)? \(n = 25\)? What are the actual error probabilities for the chosen n?

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?

A spectrophotometer used for measuring CO concentration (ppm (parts per million) by volume) is checked for accuracy by taking readings on a manufactured gas (called span gas) in which the CO concentration is very precisely controlled at \(70ppm\). If the readings suggest that the spectrophotometer is not working properly, it will have to be recalibrated. Assume that if it is properly calibrated, measured concentration for span gas samples is normally distributed. On the basis of the six readings \(85,{\rm{ }}77,{\rm{ }}82,{\rm{ }}68,{\rm{ }}72,\) and \(69\) is recalibration necessary? Carry out a test of the relevant hypotheses using a \(\alpha = .05\).

A sample of n sludge specimens is selected and the pH of each one is determined. The one-sample t test will then be used to see if there is compelling evidence for concluding that true average pH is less than 7.0. What conclusion is appropriate in each of the following situations?

a.n= 6, t= -2.3, 伪= .05

b.n= 15, t= -3.1伪=.01

c.n= 12, t= -1.3, 伪= .05

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