Respuesta :
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 [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.
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 [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 = 5850, \sigma = 1125, n = 20, s = \frac{1125}{\sqrt{20}} = 251.56[/tex]
Which of the following mean costs would be considered unusual?
We have to find the z-score for each of them
A. $6350
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{6350 - 5850}{251.56}[/tex]
[tex]Z = 1.99[/tex]
Not unusual
B. $6180
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{6180 - 5850}{251.56}[/tex]
[tex]Z = 1.31[/tex]
Not unusual
C. $5180
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{5180 - 5850}{251.56}[/tex]
[tex]Z = -2.66[/tex]
Unusual, and this is the answer.
