Question

Intelligence Quotient (IQ) scores are often reported to be normally distributed with μ=100.0

and σ=15.0
. A random sample of 41
people is taken.
What is the probability of a random person on the street having an IQ score of less than 99
? Round your answer to 4
decimal places, if necessary.

Answer 1

To find the probability of a random person having an IQ score less than 99, we can use the standard normal distribution (Z-distribution) because IQ scores are assumed to be normally distributed with a mean (\(\mu\)) of 100 and a standard deviation (\(\sigma\)) of 15.

First, calculate the z-score using the formula:

\[ Z = \frac{X - \mu}{\sigma} \]

Where \( X = 99 \), \( \mu = 100 \), and \( \sigma = 15 \). Plug in the values:

\[ Z = \frac{99 - 100}{15} = -0.0667 \]

Next, look up the z-score in a standard normal table or use a calculator to find the cumulative probability corresponding to \( Z = -0.0667 \). The probability \( P(Z < -0.0667) \) represents the probability of a random person having an IQ score less than 99.

Using a standard normal table or a calculator, the probability approximately equals 0.4721 when rounded to 4 decimal places.

So, the probability of a random person on the street having an IQ score of less than 99 is approximately \( 0.4721 \).