Respuesta :
Answer:
Step-by-step explanation:
x, height of men is N(69, 2.8)
Sample size n =150
Hence sample std dev = [tex]\frac{2.8}{\sqrt{150} } =0.229[/tex]
Hence Z score = [tex]\frac{x-69}{0.229}[/tex]
A) Prob that a random man from 150 can fit without bending
= P(X<78) = P(Z<3.214)=1.0000
B) n =75
Sample std dev = [tex]\frac{2.8}{\sqrt{75} } =0.3233[/tex]
P(X bar <72) = P(Z<9.28) = 1.00
C) Prob of B is more relevent because average male passengers would be more relevant than a single person
(D) The probability from part (b) is more relevant because it shows the proportion of flights where the mean height of the male passengers will be less than the door height.
Using the normal distribution and the central limit theorem, we have that:
a) The probability is 0.9983.
b) The probability is 1.
c) The probability of part a is more relevant, as the mean of the heights is always going to be close to the population mean, but the airline wants to avoid even a small number of passengers having trouble passing through the door.
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It measures how many standard deviations the measure is from the mean.
- After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of measure X.
In this problem:
- Mean of 69.8 inches, thus [tex]\mu = 69.8[/tex]
- Standard deviation of 2.8 inches, thus [tex]\sigma = 2.8[/tex].
Item a:
This is the probability of a height less than 78 in, which is the p-value of Z when X = 78.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{78 - 69.8}{2.8}[/tex]
[tex]Z = 2.93[/tex]
[tex]Z = 2.93[/tex] has a p-value of 0.9983.
The probability is 0.9983.
Item b:
- 75 men, thus [tex]n = 75[/tex], and by the Central Limit Theorem, [tex]s = \frac{2.8}{\sqrt{75}}[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{78 - 69.8}{\frac{2.8}{\sqrt{75}}}[/tex]
[tex]Z = 25.4[/tex]
[tex]Z = 25.4[/tex] has a p-value of 1
The probability is 1.
Item c:
The probability of part a is more relevant, as the mean of the heights is always going to be close to the population mean, but the airline wants to avoid even a small number of passengers having trouble passing through the door.
A similar problem is given at https://brainly.com/question/22934264
