Question

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 \(H_{d}: \mu>\) \(150^{\circ} \mathrm{F}\). In the context of this example, describe Type I and Type II errors. Which type of error would you consider more serious? Explain.

Short answer

A Type I error would imply that the power plant is non-compliant when it is actually compliant. A Type II error would imply that the power plant is compliant when it's actually not. In this context, a Type II error is more serious since it could lead to detrimental effects on the river's ecosystem.

Step-by-step solution

Define Type I error in this context

A Type I error in the context of this research would mean that the null hypothesis (\(H_{0}: \mu=150^{\circ} \mathrm{F}\)) which states that the mean temperature of the discharged water is at most 150 degree Fahrenheit is true yet we still reject it, suggesting that the power plant is non-compliant with regulations even when it is.

Define Type II error in this context

A Type II error in the context of this research would mean that we fail to reject the false null hypothesis (\(H_{0}: \mu=150^{\circ} \mathrm{F}\)). This would mean that the power plant is actually non-compliant, with its discharged water being above the acceptable temperature, yet the hypothesis testing indicated it was compliant.

Determining the more serious error

Given the ecosystem would be negatively impacted if the mean discharge water temperature is above 150 degrees Fahrenheit, a Type II error is considered more serious. This is because a Type II error would lead us to believe that the power plant is compliant and not causing harm, when in reality it is causing harmful ecological effects by discharging water hotter than the acceptable temperature.

Key concepts

Type I and Type II Errors

In the realm of hypothesis testing in statistics, differentiating between Type I and Type II errors is critical to understanding the consequences of erroneous conclusions. A Type I error occurs when a true null hypothesis is rejected. Imagine a scenario where a diligent student is incorrectly accused of cheating simply because their correct answers coincidentally matched those of a neighbor. Similarly, in our power plant example, a Type I error would implicate the power plant in violation of environmental temperature regulations even when it is, in fact, operating within the permissible range.

Conversely, a Type II error happens when a false null hypothesis is not rejected. This could be likened to a thief walking free due to inadequate evidence, despite committing a crime. In the context of our example, a Type II error would mean the power plant is erroneously deemed compliant while it discharges water that is too hot, posing a risk to the river's ecosystem. As we weigh the implications, it's often contended that a Type II error is more grave. Overlooking actual wrongdoing can lead to serious and lasting damage to our environment, which can be far more challenging to rectify than an unjust accusation (Type I error).

Statistical Hypothesis

A statistical hypothesis is an assumption or theory about a parameter of a population that can be tested using statistical methods. It's akin to a scientific hypothesis, which is an educated guess about the relationships among variables. In hypothesis testing, we use a null hypothesis (labeled as \( H_{0} \)) to represent the default or status quo condition, like a judge's presumption of innocence in a trial. For our example, the null hypothesis \( H_{0}: \mu=150^{\textdegree} F \) states that the power plant's discharged water temperature is safe for the ecosystem.

An alternative hypothesis (\( H_{d} \)), on the other hand, challenges this status quo, proposing an effect or difference. In the case of the power plant, the alternative hypothesis \( H_{a}: \mu>150^{\textdegree} F \) suggests that the discharged water's mean temperature does exceed the safe threshold. The goal of sampling and data analysis is to collect evidence that determines which hypothesis is more likely to be true based on observed data.

Environmental Impact of Power Plants

Power plants, particularly those using non-renewable energy sources, can have a substantial impact on the environment. This influence is not just confined to air quality but also extends to water bodies. The discharge of heated water into rivers, a process known as thermal pollution, can disturb local aquatic life, leading to ecosystem imbalances. Such ecological disturbances affect not just the temperature but also the oxygen levels in water, which can harm or kill fish and other aquatic organisms.

In our textbook example, ensuring that the mean temperature of the discharged water does not exceed \(150^{\textdegree} F\) is essential to protect the delicate balance of riverine ecosystems. Overheated water may cause species to migrate, alter their reproductive patterns, or even lead to their extinction. Thus, monitoring and managing the environmental parameters like discharge water temperature is crucial to mitigate power plants' impact on their surroundings.

Sampling and Data Analysis

Sampling and data analysis are the backbone of empirical research, providing a systematic approach for making inferences about a larger population. In the context of evaluating the environmental compliance of a power plant, sampling refers to the process of selecting a subset of water samples from the discharge to estimate the temperature of the entire output. This method is practical and cost-effective because it can be nearly impossible to measure every instance of discharge.

The data collected are then subject to data analysis, where statistical techniques are applied to interpret the numbers and arrive at a conclusion regarding the initial hypothesis. Key steps in this analysis include summarizing the data, estimating population parameters, and performing hypothesis tests. Several outcomes are possible, and as we've learned, the stakes in environmental monitoring are high, as any misinterpretation could lead to an incorrect conclusion about the power plant’s impact on the aquatic ecosystem.