Question

IQ scores (as measured by the Stanford-Binet intelligence test) are normally distributed with a mean of 100 and a standard deviation of 19.Find the approximate number of people in the United States (assuming a total population of 280,000,000) with an IQ higher than 140. (Round your answer to the nearest hundred thousand.)

Answer 1

`P(X>140)=1-P(X\leq 140)=1-P((X-\mu)/(\sigma)<(140-\mu)/(\sigma))=1-P(Z<(140-100)/(19))=1-P(Z<2.105)=0.0176`

Number =0.0176*280000000=4928000 approximately Americans would have a IQ score more than higher than 140

Explanation:

1) Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

2) Solution to the problem

Let X the random variable that represent the IQ scores of the population of interest, and for this case we know the distribution for X is given by:

`X \sim N(100,19)`

Where
`\mu=100` and
`\sigma=19`

We are interested on this probability

`P(X>140)`

And the best way to solve this problem is using the normal standard distribution and the z score given by:

`z=(x-\mu)/(\sigma)`

If we apply this formula and the complement rule to our probability we got this:

`P(X>140)=1-P(X\leq 140)=1-P((X-\mu)/(\sigma)<(140-\mu)/(\sigma))=1-P(Z<(140-100)/(19))=1-P(Z<2.105)=0.0176`

And we can find this probability with the following excel code:

"=1-NORM.DIST(2.105,0,1,TRUE)"

This number 0.0176 represent the proportion of Americans that present a score higher than 140.

And now since we ar einterested on the approximate number of people in the United States (assuming a total population of 280,000,000) with an IQ higher than 140, we just need to do this:

Number =0.0176*280000000=4928000 approximately Americans would have a IQ score more than higher than 140