Question

some psychologists believe that a genius should be defined as anyone having an IQ over 140. If IQ scores are normally distributed with a mean of 100 and a standard deviation of 17 and if the population of the world is 6,575,000,000, how many geniuses are there in the world today

Answer 1

61,239,550

Explanation:

We let the random variable X denote the IQ scores. This would imply that X is normal with a mean of 100 and standard deviation of 17. We proceed to determine the probability that an individual chosen at random from the population would be a genius, that is;

Pr( X>140)

The next step is to evaluate the z-score associated with the IQ score of 140 by standardizing the random variable X;

`Pr(X>140)=Pr(Z>(140-100)/(17))=Pr(Z>2.3529)`

The area to the right of 2.3529 will be the required probability. This area from the standard normal tables is 0.009314

From a population of 6,575,000,000 the number of geniuses would be;

6,575,000,000*0.009314 = 61,239,550

Answer 2

61,805,000

Explanation:

lets assume, no of geniuses = X

P(X > 140) = P(z > (140 - 100)/17)

= P(z > 40/17) ⇒ P(z > 2.352)

1 - P(z < 2.352) = 1 - 0.9906 ⇒ 0.0094

The number of geniuses in the world today are:

0.0094 x 6,575,000,000 = 61,805,000