Question

IQ scores are measured with a test designed so that the mean is 116 and the standard deviation is 16. Consider the group of IQ scores that are unusual. What are the z scores that separate the unusual IQ scores from those that are​ usual? What are the IQ scores that separate the unusual IQ scores from those that are​ usual? (Consider a value to be unusual if its z score is less than minus2 or greater than​ 2.)

Answer 1

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 $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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:

$\mu = 116, \sigma = 16$

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

$Z = \frac{X - \mu}{\sigma}$

$-2 = \frac{X - 116}{16}$

$X - 116 = -2*16$

$X = 84$

Z = 2

$Z = \frac{X - \mu}{\sigma}$

$2 = \frac{X - 116}{16}$

$X - 116 = 2*16$

$X = 148$

The IQ scores that separate the unusual IQ scores from those that are​ usual are 84 and 148.

Answer 2

Answer:

$\mu -2*\sigma = 116-2*16= 84$

$\mu +2*\sigma = 116+2*16= 148$

Step-by-step explanation:

We know that the distirbution for the IQ have the following parameters:

$\mu=116, \sigma=16$

The z score is given by this formula

$z = \frac{X -\mu}{\sigma}$

And replacing the limits we got:

$X = \mu \pm z \sigma$

For this case we know that a value to be unusual if its z score is less than minus2 or greater than​ 2, so then we can find the limits with this formulas:

$\mu -2*\sigma = 116-2*16= 84$

$\mu +2*\sigma = 116+2*16= 148$