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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 \(150^{\circ} \mathrm{F}\), researchers will take 50 water samples at randomly selected times and record the temperature of each sample. The resulting data will be used to test the hypotheses \(H_{0}: \mu=150^{\circ} \mathrm{F}\) versus \(\left.H_{a}: \mu\right\rangle\) \(150^{\circ} \mathrm{F}\). In the context of this example, describe Type I and Type II errors. Which type of error would you consider more serious? Explain.

Step by step solution

01

Understanding Type I Error

A Type I error would occur if the actual mean temperature of the water is at most 150 degrees Fahrenheit, but the test leads to the rejection of the null hypothesis, \(H_{0}\). This means that the researchers incorrectly conclude that the power plant is in violation of regulations when it isn't.

02

Understanding Type II Error

A Type II error would occur if the mean temperature of the discharged water is more than 150 degrees Fahrenheit, but the test lead to acceptance of the null hypothesis. This means the researchers incorrectly conclude that the power plant is in compliance with regulations when it really isn't.

03

Evaluating Which Error is More Serious

Which error is more serious depends on the context. However, considering the potential ecological damage, a Type II error could be considered more serious in this case. Overlooking a situation where the power plant discharges too hot water could have serious negative effects on the river's ecosystem, potentially killing off species and disrupting balance.

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Key Concepts

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

Type I Error

In statistical hypothesis testing, a Type I error occurs when the null hypothesis is rejected while it is actually true. Consider the situation in the context of the power plant scenario. Scientists believe that the mean temperature of discharged water should not exceed 150°F to prevent harm to the river's ecosystem. If the null hypothesis (the water temperature is 150°F or less) is mistakenly rejected, the researchers would erroneously conclude that the plant is not within safe limits despite actually being compliant.

This false alarm represents a Type I error. The consequence of making such an error includes unnecessary regulatory actions against the power plant, which could lead to financial and operational impacts without any real environmental benefit.

• Incorrectly rejecting the null hypothesis
• Concluding a violation that did not occur
• Potential needless interventions

Type II Error

A Type II error happens when the null hypothesis is not rejected when it should be. In our scenario, this would mean that the mean temperature of discharged water is actually above 150°F, but researchers incorrectly accept the null hypothesis instead of rejecting it. Essentially, it means concluding compliance with environmental regulations when there is, indeed, a breach.

This could have serious environmental consequences, as overlooking the real over-limit temperature could lead to thermal pollution. This might damage ecological balance, affecting aquatic life in the river negatively, sometimes irreversibly.

• Failing to reject the null hypothesis when it is false
• Inaccurate compliance indication
• Risk of ecological damage

Environmental Impact Assessment

Environmental Impact Assessments (EIAs) are critical in identifying and managing potential negative environmental effects of industrial activities like power generation. In the case of the power plant, ongoing temperature monitoring of discharged water forms part of such an assessment.

These assessments help ensure that industrial activities do not significantly harm ecosystems. They provide a framework for measuring the impact and implementing mitigation strategies. By assessing potential Type I and Type II errors in hypothesis testing, EIAs highlight the importance of accurate data interpretation in preserving environmental health.

• Monitor and assess potential ecological impacts
• Ensure compliance with environmental standards
• Help create sustainable operational practices