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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 and standard deviation
, the z-score of a measure X is given by:
- 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
.
In this problem, for the population, the mean and the standard deviation are given by, respectively:
.
For a sample of 37 passengers, we have that:
The probability that the aircraft is overloaded is <u>one subtracted by the p-value of Z when X = 167.6</u>, hence:
By the Central Limit Theorem:
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