Respuesta :
Answer:
The level is L = 0.084
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central limit theorem:
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
[tex]\mu = 0.08, \sigma = 0.01, n = 36, s = \frac{0.01}{\sqrt{36}} = 0.0017[/tex]
What is the level L such that the probability that the average NOX level x for the fleet is greater than L is only 0.01?
This is X when Z has a pvalue of 1-0.01 = 0.99. So it is X when Z = 2.325.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central limit theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]2.325 = \frac{X - 0.08}{0.0017}[/tex]
[tex]X - 0.08 = 2.325*0.0017[/tex]
[tex]X = 0.084[/tex]
The level is L = 0.084
