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

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

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

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 = (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`

Answer 2

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.

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 = (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 = (X - \mu)/(\sigma)`

`-2 = (X - 116)/(16)`

`X - 116 = -2*16`

`X = 84`

Z = 2

`Z = (X - \mu)/(\sigma)`

`2 = (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.