Question

The duration of Alzheimer’s disease from onset of symptoms to death ranges from 3 to 20 years with a mean of 8 years and standard deviation of 4 years. Suppose a medical administrator selects a random sample 36 diseased Alzheimer’s patients, and records the duration of each patient.

a. Give the properties of the sampling distribution for the sample mean duration.

b. Find the probability the sample mean duration will be less than 6 years.

c. Find the probability the sample mean will be within 1 year of the population mean duration?

d. If the medial administrator wanted a 99% chance the sample mean would be within one year of the population mean, how many patient records would need to be sampled?

e. Could the population of individual duration lengths have a normal distribution? Explain.

Answer 1

(a)

The sampling distribution for the sample mean duration is Normal Distribution with mean = 8 and standard deviation = 4/ = 0.6667

(b)

To find P(<6):

Z = (6 - 8)/0.6667

= - 3.00

By Technology, Cumulative Area Under Standard Normal Curve = 0.0013

So,

the probability the sample mean duration will be less than 6 years. = 0.0013

(c)

Z = 1/0.6667

= 1.50

Table of Area Under Standard Normal Curve gives area = 0.4332

So,

the probability the sample mean will be within 1 year of the population mean duration = 2 X 0.4332 = 0.8664

(d)

Sample Size (n) is given by:

Given:

= 0.01

From Table, critical values of Z = 2.576

= 4

e = 1

Substituting, we get:

So,

Answer is:

107

(e)

The the population of individual duration lengths could not have a normal distribution since it is known that it is a highly skewed distribution.