Question

A hospital has determined that the average duration of a stay at the hospital is 9.3 days. Suppose that the population standard deviation for the duration of stays at this hospital is 2.3 days. Researchers randomly select 36 former patients to include in further research on hospital stays. Use the central limit theorem to find the approximate probability that total amount of time these 36 patients spent at the hospital exceeds 180 days?

Answer 1

Let X_i denote the duration of stay (in days) of the i^th patient. (i = 1,2,...,36)

Now, we are given that:

Let X denote the sum of duration of stays of these 36 patients.

Thus, X = X₁+X₂+...+X₃₆ =

Now, consider:

Also:

Thus, from the central limit theorem, we can conclude that:

Now, the probability that total amount of time these 36 patients spent at the hospital exceeds 180 days is given by:

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