Question

Let X be the lifetime of an electronic device. It is known that the average lifetime of the device is 762 days and the standard deviation is 150 days. Let x¯ be the sample mean of the lifetimes of 156 devices. The distribution of X is unknown, however, the distribution of x¯should be approximately normal according to the Central Limit Theorem. Calculate the following probabilities using the normal approximation.

(a) P(x¯≤746)

(b) P(x¯≥782)

(c) P(738≤x¯≤781)

Answer 1

Solution:

We are given that: X be the lifetime of an electronic device with the average lifetime of the device is 762 days and the standard deviation is 150 days. Thus and

Sample size = n = 156

The sampling distribution of sample mean is approximately normal according to the Central Limit Theorem.

Thus mean of sample means is:

The standard deviation of sample means is:

Part a) Find:

Find z score:

Thus we get:

Look in z table for z = -1.3 and 0.03 and find corresponding area.

Thus from z table ,we get:

P( Z <-1.33 ) = 0.0918

Thus

Part b)

Thus

Find z score:

Thus we get:

Look in z table for z = 1.6 and 0.07 and find corresponding area.

Thus from z table ,we get:

P( Z < 1.67 ) =0.9525

Thus

Part c) .

and

Thus we get:

Look in z table for z= 1.5 and 0.08 as well as for z = -2.0 and 0.00 and find corresponding area.

Thus from z table we get:

P( Z < 1.58)= 0.9429

P( Z < -2.00)= 0.0228

Thus