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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 as a result of fuse malfunction. To verify the mean amperage of the fuses, a sample of fuses is selected and tested. If a hypothesis test is performed using the resulting data, what null and alternative hypotheses would be of interest to the manufacturer?

Short Answer

Expert verified
Null Hypothesis (H0): µ = 40 (The mean amperage of the fuses is 40 A)\nAlternative Hypothesis (H1): µ ≠ 40 (The mean amperage of the fuses is not 40 A)

Step by step solution

01

Understand the Situation

A fuse manufacturer wants to check if the mean amperage at which its fuses burn out is indeed 40 amp. If the mean amperage is lower or higher than 40 amp, there could be complaints from customers or potential liability issues. This needs to be tested statistically.
02

Formulate the Null Hypothesis

The null hypothesis (H0) is the status quo or the claim that the manufacturer wants to test. Here, it's the statement that the mean amperage of the fuses is 40 amp. So we get H0: µ = 40.
03

Formulate the Alternative Hypothesis

The alternative hypothesis (H1) is the statement against the null hypothesis. In this case, it's the statement that the mean amperage is not equal to 40 A. So, we get H1: µ ≠ 40.

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Key Concepts

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

Null Hypothesis
When it comes to hypothesis testing in statistics, the null hypothesis is a critical starting point. It represents the default statement or the current view that there is no effect or no difference. In essence, it's the assumption that any kind of variation or difference you observe in your data is merely due to chance, and not due to any actual effect.

In the case of the fuse manufacturer, the null hypothesis is set up to test whether the mean amperage of the fuses is exactly 40 amps, as claimed. So the formal statement of the null hypothesis is: \(H_0: \text{The mean amperage} = 40 \text{ amps}\). If the data we collect provides strong enough evidence to refute this hypothesis, only then we would consider alternatives.
Alternative Hypothesis
Contrasting with the null hypothesis, the alternative hypothesis signifies what we aim to demonstrate or support. This hypothesis is a statement that proposes there is a statistically significant effect or difference.

For our fuse manufacturer, the concern lies in whether the mean amperage is not actually 40 amps—it could be either lower or higher. Therefore, the alternative hypothesis is a two-sided test, which we can express as: \(H_1: \text{The mean amperage} eq 40 \text{ amps}\). Successfully proving this alternative hypothesis can lead to a change in business practices, improvements in the product, or the validation of concerns over product performance.
Mean Amperage
Mean amperage is the average current at which a batch of fuses is expected to burn out. This average value is crucial for manufacturers who need to ensure their fuses perform correctly within a safe and expected range. Think of it as ensuring the right balance — not burning out too soon (causing customer inconvenience) and not holding on too long (which could raise safety concerns).

For the discussed exercise, establishing whether the actual mean amperage aligns with the stated 40 amps is vital. It anchors the claims of the manufacturer and dictates customer satisfaction as well as legal compliance and safety.
Statistical Significance
Statistical significance is a measure of whether the effect or difference observed is likely to be due to chance or if it's a result of a specific cause. It's what determines the validity of our hypothesis test results. To establish statistical significance, we compare the p-value — the probability of obtaining an effect at least as extreme as what was observed, given that the null hypothesis is true — to a predefined significance level, often 0.05 or 5%.

If the p-value is less than the significance level, we conclude that our findings are statistically significant and consequently reject the null hypothesis. This is a pivotal moment in hypothesis testing because it can influence decisions and incite changes based on the data analysis. In the context of our example, finding statistical significance would mean that the manufacturer needs to reassess the fuse design or production process to ensure the mean amperage is consistently at the desired level.

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Most popular questions from this chapter

Optical fibers are used in telecommunications to transmit light. Suppose current technology allows production of fibers that transmit light about \(50 \mathrm{~km} .\) Researchers are trying to develop a new type of glass fiber that will increase this distance. In evaluating a new fiber, it is of interest to test \(H_{0}: \mu=50\) versus \(H_{a}: \mu>50\), with \(\mu\) denoting the mean transmission distance for the new optical fiber. a. Assuming \(\sigma=10\) and \(n=10,\) use Appendix Table 5 to find \(\beta,\) the probability of a Type II error, for each of the given alternative values of \(\mu\) when a test with significance level .05 is employed: i. 52 ii. 55 \(\begin{array}{ll}\text { iii. } 60 & \text { iv. } 70\end{array}\) b. What happens to \(\beta\) in each of the cases in Part (a) if \(\sigma\) is actually larger than 10? Explain your reasoning.

The power of a test is influenced by the sample size and the choice of significance level. a. Explain how increasing the sample size affects the power (when significance level is held fixed). b. Explain how increasing the significance level affects the power (when sample size is held fixed).

10.52 - Medical research has shown that repeated wrist extension beyond 20 degrees increases the risk of wrist and hand injuries. Each of 24 students at Cornell University used a proposed new computer mouse design, and while using the mouse, each student's wrist extension was recorded. Data consistent with summary values given in the paper "Comparative Study of Two Computer Mouse Designs" (Cornell Human Factors Laboratory Technical Report \(\mathrm{RP} 7992\) ) are given. Use these data to test the hypothesis that the mean wrist extension for people using this new mouse design is greater than 20 degrees. Are any assumptions required in order for it to be appropriate to generalize the results of your test to the population of Cornell students? To the population of all university students? \(\begin{array}{llllllllllll}27 & 28 & 24 & 26 & 27 & 25 & 25 & 24 & 24 & 24 & 25 & 28 \\ 22 & 25 & 24 & 28 & 27 & 26 & 31 & 25 & 28 & 27 & 27 & 25\end{array}\)

Occasionally, warning flares of the type contained in most automobile emergency kits fail to ignite. A consumer advocacy group wants to investigate a claim against a manufacturer of flares brought by a person who claims that the proportion of defective flares is much higher than the value of .1 claimed by the manufacturer. A large number of flares will be tested, and the results will be used to decide between \(H_{0}: p=.1\) and \(H_{a}: p>.1,\) where \(p\) represents the proportion of defective flares made by this manufacturer. If \(H_{0}\) is rejected, charges of false advertising will be filed against the manufacturer. a. Explain why the alternative hypothesis was chosen to be \(H_{a}: p>.1 .\) b. In this context, describe Type I and Type II errors, and discuss the consequences of each.

A comprehensive study conducted by the National Institute of Child Health and Human Development tracked more than 1000 children from an early age through elementary school (New york Times, November 1,2005\()\). The study concluded that children who spent more than 30 hours a week in child care before entering school tended to score higher in math and reading when they were in the third grade. The researchers cautioned that the findings should not be a cause for alarm because the effects of child care were found to be small. Explain how the difference between the sample mean math score for third graders who spent long hours in child care and the known overall mean for third graders could be small but the researchers could still reach the conclusion that the mean for the child care group is significantly higher than the overall mean for third graders.

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