Answer:
C. $5180
Step-by-step explanation:
To solve this question, we have 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
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
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.
Z-scores lower than -2 or higher than 2 are considered unusual.
Central limit theorem:
The Central Limit Theorem estabilishes that, for a random normally distributed variable X, with mean
and standard deviation
, the sample means with size n can be approximated to a normal distribution with mean
and standard deviation ![s = \frac{\sigma}{\sqrt{n}}](https://tex.z-dn.net/?f=s%20%3D%20%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D)
In this problem, we have that:
![\mu = 5850, \sigma = 1125, n = 20, s = \frac{1125}{\sqrt{20}} = 251.56](https://tex.z-dn.net/?f=%5Cmu%20%3D%205850%2C%20%5Csigma%20%3D%201125%2C%20n%20%3D%2020%2C%20s%20%3D%20%5Cfrac%7B1125%7D%7B%5Csqrt%7B20%7D%7D%20%3D%20251.56)
Which of the following mean costs would be considered unusual?
We have to find the z-score for each of them
A. $6350
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
By the Central Limit Theorem
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{6350 - 5850}{251.56}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B6350%20-%205850%7D%7B251.56%7D)
![Z = 1.99](https://tex.z-dn.net/?f=Z%20%3D%201.99)
Not unusual
B. $6180
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{6180 - 5850}{251.56}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B6180%20-%205850%7D%7B251.56%7D)
![Z = 1.31](https://tex.z-dn.net/?f=Z%20%3D%201.31)
Not unusual
C. $5180
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{5180 - 5850}{251.56}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B5180%20-%205850%7D%7B251.56%7D)
![Z = -2.66](https://tex.z-dn.net/?f=Z%20%3D%20-2.66)
Unusual, and this is the answer.