Question

An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 130 lb and 191 lb. The new population of pilots has normally distributed weights with a mean of 137 and a standard deviation of 30.8lb.

a. If a pilot is randomly​ selected, find the probability that his weight is between 130lb and 191lb?

b. If different pilots are randomly​ selected, find the probability that their mean weight is between 130lb and 191lb.

​(Round to four decimal places as​ needed.)

Answer 1

Let X be pilot weights

X is normally distributed with and

a) If a pilot is randomly​ selected, find the probability that his weight is between 130lb and 191lb?

It is asked that P(130 < X < 191), to standardise this substitute

=> P(130 < X < 191) = P( (130-137)/30.8 < Z < (191-137)/30.8 ) = P(-0.227 < Z < 1.753)

From standard normal distribution table,

P(-0.227 < Z < 1.753) = 0.96 - 0.41 = 0.55

b) If different pilots are randomly​ selected, find the probability that their mean weight is between 130lb and 191lb.

Here the question should specify the number of pilots, like "if 27 different pilots...." or "if 52 different pilots...." etc.

For now, let it be n.

The standard deviation of a sample of size n is  

Let their mean be .

We have to find

P(130 < < 191)

And here we have to substitute and the rest of the process is same like part a.

Again we will have some expression like P( a< Z < b) after this substitution which we can find out using the standard normal distribution table.