Answer:
87.49% probability that the sample average was less than 375 days
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 are solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
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
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this question, we have that:
![\mu = 359, \sigma = 90.4, n = 42, s = \frac{90.4}{\sqrt{42}} = 13.95](https://tex.z-dn.net/?f=%5Cmu%20%3D%20359%2C%20%5Csigma%20%3D%2090.4%2C%20n%20%3D%2042%2C%20s%20%3D%20%5Cfrac%7B90.4%7D%7B%5Csqrt%7B42%7D%7D%20%3D%2013.95)
What is the probability that the sample average was less than 375 days?
This is the pvalue of Z when X = 375. So
By the Central Limit Theorem
![Z = 1.15](https://tex.z-dn.net/?f=Z%20%3D%201.15)
has a pvalue of 0.8749.
87.49% probability that the sample average was less than 375 days