Answer:
Z = 1
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 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:
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.
Central Limit Theorem
The Central Limit Theorem establishes 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.
Mean (mu) that equals 100 with a standard deviation (sigma) of 18
![\mu = 100, \sigma = 18](https://tex.z-dn.net/?f=%5Cmu%20%3D%20100%2C%20%5Csigma%20%3D%2018)
Sample of 9:
This means that ![n = 9, s = \frac{18}{\sqrt{9}} = 6](https://tex.z-dn.net/?f=n%20%3D%209%2C%20s%20%3D%20%5Cfrac%7B18%7D%7B%5Csqrt%7B9%7D%7D%20%3D%206)
What will be the computed z-score with a sample mean (x-bar) of 106?
This is Z when X = 106. So
![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{106 - 100}{6}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B106%20-%20100%7D%7B6%7D)
![Z = 1](https://tex.z-dn.net/?f=Z%20%3D%201)
So Z = 1 is the answer.