Question

10. An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 150 lb and 191 lb. The new population of pilots has normally distributed weights with a mean of 156 lb and a standard deviation of 26.9 lb.

b. If 32 different pilots are randomly​ selected, find the probability that their mean weight is between 150

lb and 191 lb.

The probability is approximately _____

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

11. A boat capsized and sank in a lake. Based on an assumption of a mean weight of 140 lb, the boat was rated to carry 60 passengers​ (so the load limit was 8,400 lb). After the boat​ sank, the assumed mean weight for similar boats was changed from 140 lb to 174 lb. Complete parts a and b below.

a. Assume that a similar boat is loaded with 60 passengers, and assume that the weights of people are normally distributed with a mean of 182.2 lb and a standard deviation of 39.8 lb. Find the probability that the boat is overloaded because the 60 passengers have a mean weight greater than 140 lb.

The probability is

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

Answer 1

Solution:

Question 10)

We are given that the new population of pilots has normally distributed weights with a mean of 156 lb and a standard deviation of 26.9 lb.

That is: and

An engineer is going to redesign an ejection seat for an airplane.

The seat was designed for pilots weighing between 150 lb and 191 lb.

Part b)

Sample size = n = 32

We have to find the probability that their mean weight is between 150 lb and 191 lb.

That is:

Thus find z score :

and

Thus we get:

Look in z table for z = 7.3 and 0.06 as well as for z = -1.2 and 0.06 and find area.

Since z = 7.36 is too large z value, thus P(Z < 7.36) = 1.0000

P( Z < -1.26) = 0.1038

Thus

Thus the probability that their mean weight is between 150 lb and 191 lb is 0.8962

Question 11)

Given: the weights of people are normally distributed with a mean of 182.2 lb and a standard deviation of 39.8 lb.

Thus Mean =

Standard Deviation =

Sample size = n = 60

    We have to find the probability that the boat is overloaded because the 60 passengers have a mean weight greater than 140 lb.

That is:

Thus we get:

Since z= -8.21 is very small, so Area below z = -8.21 is approximately = 0

That is: P( Z < -8.21) = 0.0000

Thus

Thus the probability that the boat is overloaded is 1.0000