One tailed or two tailed Ok so this the question:
An administrator at a medium-sized hospital tells the board of directors that, among patients received at the Emergency room and eventually admitted to a ward, the average length of time between arriving at Emergency and being admitted to the ward is 4 hours and 15 minutes. One of the board members believes this figure is an underestimate and checks the records for a sample of 25 patients. The sample mean is 6 hours and 30 minutes. Assuming that the population standard deviation is 3 hours, and that the length of time spent in Emergency is normally distributed, use the sample data to determine whether there is sufficient evidence at the 5% level of significance to assert that the administrator's claim is an underestimate.
The first scenario is that that for the null hypothesis the mean is equals to 4hrs 15mins. For the alternative hypothesis the mean is not equals to 4hrs 15mins. So it could be less or more.
However the question says that one the board memeber thinks that it might be an UNDERESTIMATE so that means the alternative hypothesis must be higher than 4hrs 15 mins? Right? So that opens the possibility to a another scenario which is:
In the null hypothesis the mean is equals to or less than 4hrs 15mins. And in the alternative hypothesis the mean is greater than 4hrs 15mins.
So the first scenario is a two tailed test and the second scenario is one tailed test. The question asks me whether it is a one tailed test or two tailed, and there is only one correct answer. But I am not sure which one is correct. From my perspective both make sense. If someone could give me the correct answer and explain it to me it would help me out a lot.

There is only one scenario. The administrator's statement has to be taken at face value and is the null hypothesis.
$H_0: t=4.25$ hours
The board member has an alternative hypothesis.
$H_1: t>4.25$ hours
This is one-tailed.
