Question

1. A hospital determined that the average duration of a stay at the hospital is 7.1 days. Suppose that the population standard deviation for the duration of stays at this hospital is 1.9 days. Researchers randomly select 26 former patients to include in further research on hospital stays.

a) Use the central limit theorem to find the approximate probability that total amount of time these 26 patients spent at the hospital exceeds 156 days?

b) What assumptions must you make to use the central limit theorem in (a)? Are these assumptions reasonable in this case?

Answer 1

(a) Let X_i be the amount of time spent at the hospital by a patient, i=1,2,3,...

Define,

Then, Probability that total amount of time spent by the 26 patients at the hospital exceeds 156 days is given by,

Now,

By Central Limit Theorem,

Therefore,

So, the required probability is 0.9984

(b) The assumptions of Central Limit Theorem are that {X_i} are a sequence of independent and identically distributed random variables with common mean and finite variance .

In this case, the time spent at hospital by each patient is independent of the time spent at hospital by other patients and they are identically distributed with common mean of 7.1 days and standard deviation of 1.9 days. So the assumptions are reasonable.