Answer:
So the z-scores that separate the unusual IQ scores from those that are usual are Z = -2 and Z = 2.
The IQ scores that separate the unusual IQ scores from those that are usual are 84 and 148.
Step-by-step explanation:
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.
In this question, we have that:
![\mu = 116, \sigma = 16](https://tex.z-dn.net/?f=%5Cmu%20%3D%20116%2C%20%5Csigma%20%3D%2016)
What are the z scores that separate the unusual IQ scores from those that are usual?
If Z<-2 or Z > 2, the IQ score is unusual.
So the z-scores that separate the unusual IQ scores from those that are usual are Z = -2 and Z = 2.
What are the IQ scores that separate the unusual IQ scores from those that are usual?
Those IQ scores are X when Z = -2 and X when Z = 2. So
Z = -2
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![-2 = \frac{X - 116}{16}](https://tex.z-dn.net/?f=-2%20%3D%20%5Cfrac%7BX%20-%20116%7D%7B16%7D)
![X - 116 = -2*16](https://tex.z-dn.net/?f=X%20-%20116%20%3D%20-2%2A16)
![X = 84](https://tex.z-dn.net/?f=X%20%3D%2084)
Z = 2
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![2 = \frac{X - 116}{16}](https://tex.z-dn.net/?f=2%20%3D%20%5Cfrac%7BX%20-%20116%7D%7B16%7D)
![X - 116 = 2*16](https://tex.z-dn.net/?f=X%20-%20116%20%3D%202%2A16)
![X = 148](https://tex.z-dn.net/?f=X%20%3D%20148)
The IQ scores that separate the unusual IQ scores from those that are usual are 84 and 148.