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

For the hypotheses testing, the following pair should be selected: \(H_{0}: \mu = 5\) versus \(H_{a}: \mu > 5\). This is because our concern is the mercury level being higher than the safe level, not lower. For the significance level, a lower value of 0.01 is preferable. This reduces the risk of incorrectly deciding that the mercury level is not exceeding the safe level, thus minimizing potential health risks.

Step-by-step solution

Choice of Hypotheses

The task here is to make a decision about prohibiting fishing. If it is found that the mean mercury concentration in the fish is greater than 5ppm (considered to be the maximum safe concentration), the area will be closed. Therefore, testing will involve checking if the mean mercury concentration exceeds 5ppm or not. Thus:\n\nHypothesis pair: \(H_{0}: \mu = 5ppm) \(H_{a}: \mu > 5ppm.\)

Choice of Significance Level

The significance level determines the risk of rejecting the null hypothesis when it's true (Type I error). A significance level of 0.1 means a 10% risk of concluding that a difference exists when there does not (false positive), while a level of 0.01 means only a 1% risk of reaching the same conclusion. Given the potential dangers of consuming fish with high mercury levels, a lower risk of making this error seems beneficial. Therefore, a significance level of 0.01 might be more suitable.