The average length of a hospital stay in the US is 9 days with a standard deviation of 3 days. Assume the lengths of stays follow a Normal model.

Question

a.What is the probability that a randomly selected patient’s stay is less than 10 days?

b.What is the probability that the average length of stay for a random sample of 100 patients is less than 10 days?

c.Explain why the probabilities in a) and b) are not equal.

Answer 1

x is normally distributed with mean 3 and SD 9: N(3,9)

z = (x - mean)/SD is N(0,1)

a.

want to compute P(x > 10)

compute z for x = 10, mean = 3 and SD = 9:

z = (10 - 9)/3 = 0.33

P(x > 10) = P(z > 0.33) = 1 - P()z < 0.33) = 1 - 0.6293 = 0.3707

b.

want to compute P(xbar > 10) for sample size = 100

for mean z = (xbar - mean)/(SD/square root sample size) = (10 - 9)/(3/10) = 1/0.3 = 3.33

P(xbar > 10) = P(z > 3.33) = 1 - P(z < 3.33) = 1 - 0.9996 = 0.004

c. standard error of the mean (SD/10) is less than standard deviation of sample (SD).