Questions: Find the area of the shaded region. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15. The area of the shaded region is . (Round to four decimal places as needed.)

Solution Steps

The IQ scores of adults are normally distributed with a mean (\( \mu \)) of 100 and a standard deviation (\( \sigma \)) of 15. We are interested in finding the probability of an IQ score falling between 85 and 115.

To find the probability, we first calculate the Z-scores for the lower and upper bounds of the range:

• For the lower bound \( x = 85 \): \[ Z_{start} = \frac{85 - 100}{15} = -1.0 \]

• For the upper bound \( x = 115 \): \[ Z_{end} = \frac{115 - 100}{15} = 1.0 \]

Using the Z-scores, we can find the probability that an IQ score falls between 85 and 115 using the cumulative distribution function \( \Phi \):

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.0) - \Phi(-1.0) \]

From standard normal distribution tables or calculations, we find: \[ \Phi(1.0) \approx 0.8413 \quad \text{and} \quad \Phi(-1.0) \approx 0.1587 \]

Thus, the probability is: \[ P = 0.8413 - 0.1587 = 0.6826 \]

Final Answer