To find the probability that a randomly selected adult has an IQ less than 128, we can use the properties of the normal distribution.
Given:
- Mean (\(\mu\)) = 100
- Standard deviation (\(\sigma\)) = 20
- IQ score = 128
We will first convert the IQ score to a z-score using the following formula:
\[ z = \frac{(X - \mu)}{\sigma} \]
Where:
- \(X\) is the value we are interested in (128 in this case),
- \(\mu\) is the mean,
- \(\sigma\) is the standard deviation.
Substituting the values:
\[ z = \frac{(128 - 100)}{20} = \frac{28}{20} = 1.4 \]
Next, we will look up the z-score of 1.4 in the standard normal distribution table or use a calculator to find the corresponding probability.
Using a standard normal distribution table or calculator, we find that:
\[ P(Z < 1.4) \approx 0.9192 \]
Thus, the probability that a randomly selected adult has an IQ less than 128 is approximately 0.9192, or 91.92%.