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Chapter 9: Problem 6

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 null hypothesis is \( H_0: \mu = 40 \); the alternative hypothesis is \( H_a: \mu \neq 40 \). Type I error: reject \( H_0 \) when true; Type II error: fail to reject \( H_0 \) when false.

Step by step solution

01

Formulate the Null Hypothesis

The null hypothesis (H_0) is a statement of no effect or no difference. In our scenario, the manufacturer wants to ensure that the mean amperage is 40. Therefore, the null hypothesis will be that the mean amperage is equal to 40. Mathematically, this is represented as \( H_0: \mu = 40 \).
02

Formulate the Alternative Hypothesis

The alternative hypothesis (H_a) is the statement you want to test against the null. Since the manufacturer wants to detect if the mean differs from 40, a two-tailed test is appropriate. Therefore, the alternative hypothesis will be that the mean amperage is not equal to 40. This is represented as \( H_a: \mu eq 40 \).
03

Define Type I Error

A Type I error occurs when you reject the null hypothesis when it is actually true. In the context of this problem, it would mean concluding that the mean amperage is not 40 when, in reality, it is 40. This could lead to unnecessary modifications in the fuse production process.
04

Define Type II Error

A Type II error occurs when you fail to reject the null hypothesis when the alternative hypothesis is true. Here, it would mean concluding that the mean amperage is 40 when it actually is not. This could lead to customer complaints or potential liability issues due to inaccurate fuse amperage.

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Key Concepts

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

Null Hypothesis
In hypothesis testing, the null hypothesis is the foundation of your scientific investigation. It's the statement that suggests there is no change or no effect, serving as a default or "no surprise" scenario. For the fuse manufacturer, this involves stating that the mean amperage at which fuses burn out is exactly 40 amps. This assumption helps establish that the production process is operating as intended. We express this mathematically as \( H_0: \mu = 40 \), where \( \mu \) represents the true mean amperage. The null hypothesis is crucial because it sets the stage for establishing whether there is enough evidence to suggest deviations from what is expected or "normal." By setting \( H_0 \), the manufacturer can verify if their process needs adjustments, or if it meets predetermined quality standards.
Alternative Hypothesis
The alternative hypothesis represents what you aim to test. It reflects the possibility of a change or difference and challenges the null hypothesis. For the fuse manufacturer, the concern lies in ensuring that the mean amperage is not precisely 40 amps. A deviation could lead to customer dissatisfaction or legal responsibility. Hence, the alternative hypothesis is stated as \( H_a: \mu eq 40 \), indicating that there is a significant difference either above or below the 40-amp mark. Choosing a two-tailed test allows checking both directions of deviation. This hypothesis helps guide the analysis by focusing on detecting real changes in fuse performance and ensuring that the manufacturing process remains reliable and safe.
Type I Error
A Type I error occurs when you incorrectly reject the null hypothesis when it is indeed true. In the context of the fuse manufacturer, making a Type I error means concluding that the mean amperage deviates from 40 amps, while in reality, it is still precisely 40 amps. The consequence of a Type I error in this scenario might include unnecessary changes to the production line, increased costs due to unnecessary corrections, or an overall disruption in the manufacturing process. The risk of making a Type I error is controlled by setting a significance level in hypothesis testing, typically denoted by \( \alpha \). Understanding the balance between risk and cost is crucial when considering adjustments based on such test outcomes.
Type II Error
Type II error happens when you fail to reject the null hypothesis when it is false. This means, for our fuse manufacturer, mistakenly accepting that the mean amperage is 40 amps when it's not. The main risk here is that actual and potentially serious issues in the fuse performance are overlooked. Customers could experience frequent replacements if the amperage is less than 40, or damaging overcurrents if it's higher, leading to potential liabilities. This risk is quantified by \( \beta \), the probability of a Type II error. Ensuring a well-designed test process can minimize the chance of missing a true effect, ultimately preserving product quality and customer satisfaction.

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Most popular questions from this chapter

Show that for any \(\Delta>0\), when the population distribution is normal and \(\sigma\) is known, the twotailed test satisfies \(\beta\left(\mu_{0}-\Delta\right)=\beta\left(\mu_{0}+\Delta\right)\), so that \(\beta\left(\mu^{\prime}\right)\) is symmetric about \(\mu_{0}\).

To test the ability of auto mechanics to identify simple engine problems, an automobile with a single such problem was taken in turn to 72 different car repair facilities. Only 42 of the 72 mechanics who worked on the car correctly identified the problem. Does this strongly indicate that the true proportion of mechanics who could identify this problem is less than \(.75\) ? Compute the \(P\)-value and reach a conclusion accordingly.

Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most \(150^{\circ} \mathrm{F}\), there will be no negative effects on the river's ecosystem. To investigate whether the plant is in compliance with regulations that prohibit a mean discharge-water temperature above \(150^{\circ}, 50\) water samples will be taken at randomly selected times, and the temperature of each sample recorded. The resulting data will be used to test the hypotheses \(H_{0}: \mu=150^{\circ}\) versus \(H_{\mathrm{a}}: \mu>150^{\circ}\). In the context of this situation, describe type I and type II errors. Which type of error would you consider more serious? Explain.

A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at 40 mph under specified conditions is known to be \(120 \mathrm{ft}\). It is proposed that the new design be implemented only if sample data strongly indicates a reduction in true average braking distance for the new design. a. Define the parameter of interest and state the relevant hypotheses. b. Suppose braking distance for the new system is normally distributed with \(\sigma=10\). Let \(\bar{X}\) denote the sample average braking distance for a random sample of 36 observations. Which of the following rejection regions is appropriate: \(R_{1}=\\{\bar{x}: \bar{x} \geq 124.80\\}, R_{2}=\) \(\\{\bar{x}: \bar{x} \leq 115.20\\}, R_{3}=\\{\bar{x}:\) either \(\bar{x} \geq 125.13\) or \(\bar{x} \leq 114.87\\} ?\) c. What is the significance level for the appropriate region of part (b)? How would you change the region to obtain a test with \(\alpha=.001\) ? d. What is the probability that the new design is not implemented when its true average braking distance is actually \(115 \mathrm{ft}\) and the appropriate region from part (b) is used? e. Let \(Z=(\bar{X}-120) /(\sigma / \sqrt{n})\). What is the significance level for the rejection region \(\\{z\) : \(z \leq-2.33\\}\) ? For the region \(\\{z: z \leq-2.88\\}\) ?

Give as much information as you can about the \(P\)-value of a \(t\) test in each of the following situations: a. Upper-tailed test, \(\mathrm{df}=8, t=2.0\) b. Lower-tailed test, \(\mathrm{df}=11, t=-2.4\) c. Two-tailed test, \(\mathrm{df}=15, t=-1.6\) d. Upper-tailed test, df \(=19, t=-.4\) e. Upper-tailed test, df \(=5, t=5.0\) f. Two-tailed test, df \(=40, t=-4.8\)
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