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 \(H_{a}: \mu>150^{\circ} \mathrm{F}\). 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 be to incorrectly find that the mean temperature of the discharged water is more than 150°F, while a Type II error would be to not detect that the mean water temperature is more than 150°F when it actually is. Given the potential severe negative impact on the river's ecosystem associated with a Type II error, it would be considered more serious in this context.

Step-by-step solution

Understanding Type I and Type II errors

In statistical hypothesis testing, a Type I error occurs when the null hypothesis is rejected when it is true, while a Type II error occurs when the null hypothesis is not rejected when it is false. In this case, Type I error would mean incorrectly concluding that the temperature is more than 150 degrees when it is not, and Type II error would mean incorrectly concluding that the temperature is not more than 150 degrees when it really is.

Analyzing the implications of Type I and Type II errors

A Type I error in this scenario would raise unnecessary alarms as power plant's discharge water would be falsely marked as a threat to the river's ecosystem. On the other hand, a Type II error means that the power plant's discharge water's harmful effects are being overlooked which can result in severe damage to river's ecosystem.

Determine which type of error is more serious

In this context, a Type II error is more serious because it means the power plant's discharge water's temperature is more than 150 degrees and it is going unnoticed. This could lead to severe damage to the river's ecosystem, which is a much higher cost than raising false alarms (Type I error).