Using the <em>normal probability distribution and the central limit theorem</em>, it is found that the probability is of 0.9974 = 99.74%, which means that the pilot should take action.

<h3>Normal Probability Distribution</h3>

In a normal distribution with mean $\mu$ and standard deviation $\sigma$ , the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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, the sampling distribution of sample means of size n has standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

In this problem, for the population, the mean and the standard deviation are given by, respectively:

$\mu = 185.47, \sigma = 39$ .

For a sample of 37 passengers, we have that:

$n = 37, s = \frac{39}{\sqrt{37}} = 6.4116$

The probability that the aircraft is overloaded is <u>one subtracted by the p-value of Z when X = 167.6</u>, hence:

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem:

$Z = \frac{X - \mu}{s}$

$Z = \frac{167.6 - 185.47}{6.4116}$

$Z = -2.79$

$Z = -2.79$ has a p-value of 0.0026.

1 - 0.0026 = 0.9974.

There is a 0.9974 = 99.74% probability that the aircraft is overloaded. Since this is a very high probability, the pilot should take action.

To lern more about the <em>normal probability distribution and the central limit theorem</em>, you can check brainly.com/question/24663213

4 0

2 years ago

A colony of ants carried away 16 of your muffins. That was 4/5 of all of them! With how many did you start?

Murrr4er [49]

20 because 4/5 is 16 so 4•5=20 like 4•4=16

4 0

4 years ago

-4 + x + 7 + 3x + 6x​

-4 + x + 7 + 3x + 6x