A hotel installs smoke detectors with adjustable sensitivity in all public guest rooms. (a) State the
null and alternative hypotheses. (b) Define Type I and II error. What are the consequences of each?
(c) Which is more to be feared, and by whom? (d) If the hotel decides to reduce â risk, what would
be the consequences? Who would be affected?
I'll give you some background on Type I and Type II errors and let you take it from there.
Type I errors result when you reject the null and it's true. Type II errors result when you accept the null and it's false. You can reduce Type I errors by setting the alpha at a lower level, for example, from .05 to .01. However, when you do that, you increase the probability of making a Type II error. You would have to determine if the hotel would be more concerned about Type I or Type II errors.
(a) Null and alternative hypotheses:
Null hypothesis (H0): The smoke detectors have the same sensitivity level in all public guest rooms.
Alternative hypothesis (H1): The smoke detectors have different sensitivity levels in at least one public guest room.
(b) Type I and Type II errors:
Type I error: Type I error occurs when the null hypothesis (H0) is rejected when it is actually true. In this case, it means concluding that the smoke detectors have different sensitivity levels when they actually have the same sensitivity levels. The consequence of a Type I error is that the hotel may unnecessarily invest in adjusting the sensitivity of the smoke detectors in all public guest rooms, leading to additional costs and resources being spent.
Type II error: Type II error occurs when the null hypothesis (H0) is not rejected when it is actually false. In this case, it means failing to conclude that the smoke detectors have different sensitivity levels when they actually have different sensitivity levels. The consequence of a Type II error is that the hotel may fail to identify and address rooms with smoke detectors that require adjustment. This could potentially lead to undetected fire hazards and compromise the safety of the guests.
(c) Which is more to be feared, and by whom:
The fear of Type I and Type II errors depends on the context and the consequences of each error. In this case, the consequences of a Type I error could include unnecessary expenses for the hotel, while a Type II error may result in potential safety risks for the guests. Therefore, the hotel might fear Type II errors more as they have a direct impact on the safety of the guests.
(d) If the hotel decides to reduce the risk:
If the hotel decides to reduce the risk, they would prioritize minimizing the chance of a Type II error. This means ensuring that they accurately identify any public guest rooms with smoke detectors that require adjustment and take appropriate action to maintain the safety of the guests. By reducing the risk, the hotel can provide a safer environment for its guests.
The consequences of reducing the risk would involve potential changes to the maintenance and inspection procedures for smoke detectors, additional training for staff responsible for ensuring their proper function, and possibly some financial costs in terms of adjusting the sensitivity of the detectors.
The people affected would primarily include the hotel management, staff members responsible for maintenance and safety, and the guests staying in the public guest rooms. Ensuring their safety would be paramount, while the hotel's reputation and liability could also be affected.