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?

Note: Please metion (symboles used in the answer) what each symbole reprsents. Thanks

Answer 1

According to the central limit theorem:

If X1, X2,...Xn are independent and identically distributed random variables with finite mean $\mu$ and variance $\sigma^2$ , then the distribution of

approaches standard normal N(0,1) as n tends to infinity.

(X - )

(a)

We are provided:

My = 7.1

07 = 1.9

ny = 26

Hence the distribution of

x^ Normal(26147, 2603)

i.e.

> x~Normal(184.6, 9.6882)

The following information has been provided:

A = 184.6, 0 = 9.688137076

We need to compute

PrX > 156)

The corresponding z-value needed to be computed is:

Z-X - o 156 – 184.6 9.688137076 = -2.9521

Therefore, we get that

156 – 184.6 Pr(X > 156) = Prz>; ) = Pr(Z > -2.9521) 9.688137076)

=1- 0.0016 = 0.9984

The following is obtained graphically:

$normaldistributiongrapher.php?mean=184.6$

(b)

Assumptions must you make to use the central limit theorem:

• The underlying random variables must be independent and identically distributed
• 'n' should be large, preferably greater than 30 (depends upon the shape of the distribution of Xi's, in particular, if it is symmetrical or skewed. If symmetrical than n>10 is large enough, if skewed then n>50)

(a)

The assumptions are valid in this scenario since the hospital stay of one participant is independent of the hospital stay time of another patient. So the random variables are iid. The sample size of 26 close enough to 30 for us to use the CLT.

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