probalility question

[shaz]

The duration of Alzheirmer's disease from the onset of symptoms until death rangers from 3 to 20 years. The average is 8 years with a population deviation of 4 years. The administrator of a large medical center randomly selects the medical records of 90 deceased Alzheimer's patients from the medical center's database and records the average duration.

How would i find the probability that the average duration is less than 7 years. More than 7 years. And the average duration lies within 1 year of the population mean mu=8.

[jahred]

you aren't given any probabilistic information on the records themselves, but you do know that the sampling distribution of the sample mean is approximately normal for any reasonably behaved underlying distribution.

the individual cases themselves are distributed with a mean of 8 and standard deviation of 4, so the sample mean is normally distributed with a mean of 8 and a standard deviation of 4/sqrt(90). or,

Xbar ~ N(8, 4/sqrt(90))

now you can find the probability of these events.

i) prob. that the average duration is less than seven years

P(Xbar < 7)
= P(Z < (7-8)/[4/sqrt(90)])
= P(Z < -.4216)

ii) prob. that the average deviation is over 7 years

use a complement rule:
P(Xbar > 7) = 1 - P(Xbar < 7)

iii) prob. that the average deviation is within 1 year
P(7 < Xbar < 9)

and you can split this up as:
= P(Xbar < 9) - P(Xbar < 7)