Question

The IQ scores of a population are normally distributed with a mean of 100 and a standard deviation of 15. What percentage of the population would be recognized as having an IQ score between 115 and 130? Round to the nearest tenth. (1 point) 15.9% 35.5% 84.1% 9.9%

Answer 1

To find the percentage of the population with IQ scores between 115 and 130 in a normally distributed population with a mean of 100 and a standard deviation of 15, we can use the properties of the normal distribution.

1. **Calculate the Z-scores for both IQ scores**:
- For an IQ of 115:
\[
Z = \frac{X - \mu}{\sigma} = \frac{115 - 100}{15} = \frac{15}{15} = 1
\]

- For an IQ of 130:
\[
Z = \frac{X - \mu}{\sigma} = \frac{130 - 100}{15} = \frac{30}{15} = 2
\]

2. **Use the Z-scores to find the corresponding percentiles**:
- Using the standard normal distribution table or a calculator:
- The percentile for \( Z = 1 \) is approximately 0.8413 (or 84.13%).
- The percentile for \( Z = 2 \) is approximately 0.9772 (or 97.72%).

3. **Calculate the percentage between the two Z-scores**:
- To find the percentage of the population having an IQ between 115 and 130:
\[
P(115 < X < 130) = P(Z < 2) - P(Z < 1) = 0.9772 - 0.8413 = 0.1359
\]

4. **Convert to percentage**:
- \( 0.1359 \) as a percentage is \( 13.59\% \).

Since we need to choose the closest answer from the options provided, the percentage falls closest to **15.9%**.

Thus, the answer is **15.9%**.