Answer:
![X = 3.4161](https://tex.z-dn.net/?f=X%20%3D%203.4161)
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:
![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.
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 problem, we have that:
![\mu = 3.25, \sigma = 1.75, n = 300, s = \frac{1.75}{\sqrt{300}} = 0.101](https://tex.z-dn.net/?f=%5Cmu%20%3D%203.25%2C%20%5Csigma%20%3D%201.75%2C%20n%20%3D%20300%2C%20s%20%3D%20%5Cfrac%7B1.75%7D%7B%5Csqrt%7B300%7D%7D%20%3D%200.101)
At what value for the sample mean would be greater than exactly 95% of possible sample means?
This is X when Z has a pvalue of 0.95. So X when Z = 1.645.
![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)
![1.645 = \frac{X - 3.25}{0.101}](https://tex.z-dn.net/?f=1.645%20%3D%20%5Cfrac%7BX%20-%203.25%7D%7B0.101%7D)
![X - 3.25 = 1.645*0.101](https://tex.z-dn.net/?f=X%20-%203.25%20%3D%201.645%2A0.101)
![X = 3.4161](https://tex.z-dn.net/?f=X%20%3D%203.4161)