Question

Suppose that you are an inspector for the Fish and Game Department and that you are given the task of determining whether to prohibit fishing along part of the Oregon coast. You will close an area to fishing if it is determined that fish in that region have an unacceptably high mercury content. a. Assuming that a mercury concentration of \(5 \mathrm{ppm}\) is considered the maximum safe concentration, which of the following pairs of hypotheses would you test: $$ H_{0}: \mu=5 \text { versus } H_{a}: \mu>5 $$ or $$ H_{0}: \mu=5 \text { versus } H_{a}: \mu<5 $$ Give the reasons for your choice. b. Would you prefer a significance level of .1 or .01 for your test? Explain.

Short answer

The suitable hypotheses would be \(H_{0}: \mu=5\) and \(H_{a}: \mu>5\) because we're trying to find evidence for mercury levels above the maximum safe limit. The preferred significance level would be .01 as it reduces the risk of allowing unsafe fishing to continue.

Step-by-step solution

Choose the suitable pair of hypotheses

The null hypothesis \(H_{0}\) is generally the hypothesis that sample observations result purely from chance. In this case, that would mean that the mercury concentration in the fish population would be equal to the maximum safe concentration, which is \(5 ppm\). The alternate hypothesis \(H_{a}\) should then be the claim we are trying to prove, that the fish in this area have a mercury concentration higher than 5 ppm. So the correct set of hypotheses would be: \(H_{0}: \mu=5 \) versus \(H_{a}: \mu>5 \). This is because we're looking to find evidence for unsafe levels of mercury (above 5 ppm), not for safe levels.

Choose the significance level

The significance level depends on how much risk of making a mistake we’re willing to tolerate. Here, making a mistake would mean unnecessarily prohibiting fishing or allowing it when it poses a health risk. A lower significance level (.01) suggests a higher standard for the mercury concentration and would reduce the risk of announcing an area safe when it is not, hence it seems like the more responsible choice.

Key concepts

Null Hypothesis

In statistical hypothesis testing, the null hypothesis (\(H_{0}\)) is a statement that signifies no effect or no difference, and it proposes that any observed differences are due to random chance. In the context of mercury contamination in fish, the null hypothesis would be that the average mercury concentration in fish is at the maximum acceptable level of 5 ppm (\(\mu = 5\)). We assume that unless there is strong evidence to suggest otherwise, the mercury content is at a safe concentration.

• The null hypothesis is the default position.
• It sets the baseline that nothing unusual is happening.

Testing against this hypothesis allows us to try to disprove it, thus potentially discovering a significant problem that needs addressing.

Alternative Hypothesis

The alternative hypothesis (\(H_{a}\)) is a statement that suggests a new effect or a difference, counter to the null hypothesis. In our mercury contamination scenario, the alternative hypothesis posits that the mercury levels are higher than the safe level, specifically expressed as \(\mu > 5\) ppm. This hypothesis is what you gather evidence for within your statistical test.

• The goal is to provide evidence that supports the alternative hypothesis.
• If the evidence is strong enough to refute the null hypothesis, the alternative hypothesis may be accepted.

So essentially, any test that rejects the null hypothesis in favor of this asserts that fishing should be prohibited due to unsafe mercury levels.

Significance Level

The significance level, often denoted as \(\alpha\), is the threshold for deciding whether a given effect is statistically significant. It represents the probability of rejecting the null hypothesis when it is true, also known as a Type I error. Common values for significance levels are 0.1 or 0.01, with the latter being more stringent.

• A significance level of 0.01 means we're only willing to accept a 1% chance of mistakenly stopping fishing when it’s actually safe.
• This lower level is generally chosen in order to minimize the risk of claiming an area is safe when it could actually be harmful to continue fishing.

This demonstrates the trade-off between sensitivity to changes (finding that contamination is high when indeed it is) and protecting against false alarms that would unnecessarily prevent fishing.

Mercury Contamination

Mercury contamination in aquatic environments can occur due to various factors like industrial pollution and bioaccumulation in the food chain. This is a serious environmental and health concern as mercury is toxic and poses risks to both wildlife and humans who consume affected fish. Fish with mercury levels above safe limits can lead to neurological and developmental problems in humans.

• Safe concentration levels are determined based on extensive health studies.
• Monitoring mercury levels helps prevent health outbreaks.
• Statistical testing for mercury allows authorities to enforce regulations to protect public health and the environment.

Understanding how mercury concentration is measured and reported is vital for making informed decisions about fishing regulations and consumer safety.

Environmental Statistics

Environmental statistics involve collecting, analyzing, and interpreting data related to natural resources and ecological phenomena. It's crucial for tasks such as monitoring pollution, assessing ecosystem health, and making policy decisions. In the case of mercury contamination in fish, these statistics offer the data needed to understand contamination levels and their potential impact.

• Environmental statistics support the creation of sustainable policies.
• They serve as a tool for evaluating the success of environmental regulations.
• Help in maintaining biodiversity by ensuring species safety.

Using environmental statistics, authorities can make more informed decisions about measures like fishing bans, ensuring the ongoing protection of both human health and the environment.