Answer:
a. 341.902.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.
n instances of a normal variable:
For n instances of a normal variable, the mean is
and the standard deviation is ![s = \sigma\sqrt{n}](https://tex.z-dn.net/?f=s%20%3D%20%5Csigma%5Csqrt%7Bn%7D)
60 days, for each day, mean 6, variance of 12.
So
![\mu = 60*6 = 360](https://tex.z-dn.net/?f=%5Cmu%20%3D%2060%2A6%20%3D%20360)
![s = \sqrt{12}\sqrt{60} = 26.8328](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7B12%7D%5Csqrt%7B60%7D%20%3D%2026.8328)
What is the 25th percentile of her total wait time over the course of 60 days?
X when Z has a p-value of 0.25, so X when Z = -0.675.
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![-0.675 = \frac{X - 360}{26.8328}](https://tex.z-dn.net/?f=-0.675%20%3D%20%5Cfrac%7BX%20-%20360%7D%7B26.8328%7D)
![X - 360 = -0.675*26.8328](https://tex.z-dn.net/?f=X%20-%20360%20%3D%20-0.675%2A26.8328)
![X = 341.902](https://tex.z-dn.net/?f=X%20%3D%20341.902)
Thus, the correct answer is given by option A.