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An airliner carries 200 200 passengers and has doors with a height of 70 70 in. Heights of men are normally distributed with a mean of 69.0 69.0 in and a standard deviation of 2.8 2.8 in. Complete parts​ (a) through​ (d). a. If a male passenger is randomly​ selected, find the probability that he can fit through the doorway without bending. The probability is . 6395 . ​(Round to four decimal places as​ needed.)b. If half of the 200 200 passengers are​ men, find the probability that the mean height of the 100 100 men is less than 70 70 in. The probability is nothing . ​(Round to four decimal places as​ needed.) c. When considering the comfort and safety of​ passengers, which result is more​ relevant: the probability from part​ (a) or the probability from part​ (b)? Why? A. The probability from part​ (a) 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. B. 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. C. The probability from part​ (a) is more relevant because it shows the proportion of male passengers that will not need to bend. D. The probability from part​ (b) is more relevant because it shows the proportion of male passengers that will not need to bend. d. When considering the comfort and safety of​ passengers, why are women ignored in this​ case? A. There is no adequate reason to ignore women. A separate statistical analysis should be carried out for the case of women. B. Since men are generally taller than​ women, a design that accommodates a suitable proportion of men will necessarily accommodate a greater proportion of women. C. Since men are generally taller than​ women, it is more difficult for them to bend when entering the aircraft.​ Therefore, it is more important that men not have to bend than it is important that women not have to bend.

Respuesta :

Answer:

a) 0.6406; b) 1.000; c) C. The probability from part​ (a) is more relevant because it shows the proportion of male passengers that will not need to bend; d) B. Since men are generally taller than​ women, a design that accommodates a suitable proportion of men will necessarily accommodate a greater proportion of women.

Step-by-step explanation:

For part a,

We use the formula for the z score of an individual,

[tex]z=\frac{X-\mu}{\sigma}[/tex]

Our mean, μ, is 69 and our standard deviation, σ, is 2.8.  To find P(X < 70),

z = (70-69)/2.8 = 1/2.8 = 0.36

Using a z table, we see that the area under the curve to the left of this value is 0.6406.  This is our probability.

For part b,

We use the formula for the z score of a sample,

[tex]z=\frac{\bar{X}-\mu}{\sigma \div \sqrt{n}}[/tex]

Our mean is still 69 and our standard deviation is still 2.8.  Our sample size, n, is 100.  To find P(X < 70),

z = (70-69)/(2.8÷√100) = 1/(2.8÷10) = 1/0.28 = 3.57

Using a z table, we see the area under the curve to the left of this is greater than 0.9999.  This means that the probability is 1.000.

For part c,

We want to know the proportion of male passengers that will not need to bend.  This is why the value from part a is more important.

For part d,

Men taller on average than women.  This means if we accommodate for the mean, it follows that the women will fit as well.

Using the normal distribution and the central limit theorem, it is found that:

a) The probability is 0.6395.

b) The probability is 1.

c)

The probability from part​ (a) is more relevant because it shows the proportion of male passengers that will not need to bend.

d)

B. Since men are generally taller than​ women, a design that accommodates a suitable proportion of men will necessarily accommodate a greater proportion of women.

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 X.
  • By the Central Limit Theorem, for the sampling distribution of samples of size n, the standard deviation is [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]

In this problem:

  • Mean of 69 in, thus [tex]\mu = 69[/tex].
  • Standard deviation of 2.8 in, thus [tex]\sigma = 2.8[/tex].

Item a:

This probability is the p-value of Z when X = 70, thus:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{70 - 69}{2.8}[/tex]

[tex]Z = 0.357[/tex]

[tex]Z = 0.357[/tex] has a p-value of 0.6395.

Thus, the probability is 0.6395.

Item b:

Samples of size 100, thus [tex]n = 100, s = \frac{2.8}{\sqrt{100}} = 0.28[/tex].

This probability is the p-value of Z when X = 70, thus:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem:

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{70 - 69}{0.28}[/tex]

[tex]Z = 3.57[/tex]

[tex]Z = 3.57[/tex] has a p-value of 1.

The probability is 1.

Item c:

We want don't want many people to have to bend, thus, part a has to be considered.

Item d:

Generally, men are taller, so if the design accommodates the men, it will accommodate the women, and the correct option is:

B. Since men are generally taller than​ women, a design that accommodates a suitable proportion of men will necessarily accommodate a greater proportion of women.

A similar problem is given at https://brainly.com/question/24663213

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