word problem statistics

Question

An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 150lb and 201lb. The new population of pilots has normally distributed weights with a mean of 158lb and a standard deviation of 34.2 lb.

a. If a pilot is randomly​ selected, find the probability that his weight is between 150lb and 201lb.

b. if the number different pilots are randomly selected, find the probability that their mean weight is between 150lb and 201lb

c. When redesigning the ejection seat, which probability is more relevent ?

Answer 1

a) Let x = weight of one pilot.

Want to find P(150 < x < 201)

Since z = (x - mean) / standard deviation

= P ((150 - 158)/34.2 < (x-mean/standard deviation) < (201 - 158/34.2))

= P(-0.234 < z < 1.257)

We can now use normal probability table to find P(-0.234 < z < 1.257)

The probabilities in the normal probability table are cumulative (corresponding to less than a z value), so

the above probability is equal to:

P(z < 1.257) - P(z < -0.234) = 0.8962 - 0.409 0.4872

b) For this part we will be using Xbar (the mean) instead of X and will be using the standard error of the mean (standard deviation/square root (sample size)) in the computation of the z-values. You have not given the sample size in the statement of your problem.

c) the probability of the mean is more relevant since it gives you a better estimate.