Question

An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 120 lb and 161 lb. The new population of pilots has normally distributed weights with a mean of 129 lb and a standard deviation of 32.7 lb. a. If a pilot is randomly​ selected, find the probability that his weight is between 120 lb and 161lb. The probability is approximately _____ (Round to four decimal places as​ needed.) b. If 32 different pilots are randomly​ selected, find the probability that their mean weight is between 120 lb and 161 lb. The probability is approximately ____. ​(Round to four decimal places as​ needed.) c. When redesigning the ejection​ seat, which probability is more​ relevant?

Answer 1

a)
Here, μ = 129, σ = 32.7, x1 = 120 and x2 = 161. We need to compute P(120<= X <= 161). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (120 - 129)/32.7 = -0.28
z2 = (161 - 129)/32.7 = 0.98

Therefore, we get
P(120 <= X <= 161) = P((161 - 129)/32.7) <= z <= (161 - 129)/32.7)
= P(-0.28 <= z <= 0.98) = P(z <= 0.98) - P(z <= -0.28)
= 0.8365 - 0.3897
= 0.4468

b)
Here, μ = 129, σ = 32.7/sqrt(32) = 5.7806, x1 = 120 and x2 = 161. We need to compute P(120<= X <= 161). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (120 - 129)/5.7806 = -1.56
z2 = (161 - 129)/5.7806 = 5.54

Therefore, we get
P(120 <= X <= 161) = P((161 - 129)/5.7806) <= z <= (161 - 129)/5.7806)
= P(-1.56 <= z <= 5.54) = P(z <= 5.54) - P(z <= -1.56)
= 1 - 0.0594
= 0.9406

c)
probability from part b is more relevant