Questions: Assume that adults have IQ scores that are normally distributed with a mean of μ=105 and a standard deviation σ=15. Find the probability that a randomly selected adult has an IQ less than 126. The probability that a randomly selected adult has an IQ less than 126 is (Type an integer or decimal rounded to four decimal places as needed.)

Solution Steps

We are given that the IQ scores of adults are normally distributed with a mean $\mu = 105$ and a standard deviation $\sigma = 15$.

To find the probability that a randomly selected adult has an IQ less than 126, we first calculate the Z-score for $X = 126$ using the formula:

$$Z = \frac{X - \mu}{\sigma}$$

Substituting the values:

$$Z_{end} = \frac{126 - 105}{15} = \frac{21}{15} = 1.4$$

The probability that a randomly selected adult has an IQ less than 126 can be expressed as:

$$P(X < 126) = \Phi(Z_{end}) - \Phi(Z_{start})$$

Where $Z_{start}$ approaches negative infinity. Thus, we have:

$$P(X < 126) = \Phi(1.4) - \Phi(-\infty) = \Phi(1.4)$$

Using the standard normal distribution table or a calculator, we find:

$$\Phi(1.4) \approx 0.9192$$

Final Answer