Assume that adults have IQ scores that are normally distributed with a mean of 104 and a standard deviation of 24.5. Find the probability that a randomly selected adult has an IQ greater than 141.4.

To find the probability that a randomly selected adult has an IQ greater than 141.4, given that IQ scores are normally distributed with a mean \( \mu = 104 \) and a standard deviation \( \sigma = 24.5 \), we can follow these steps:

1. **Standardize the value 141.4** by converting it to a z-score:
   \[
   z = \frac{X - \mu}{\sigma}
   \]
   where:
   - \( X = 141.4 \) (the IQ score we're interested in),
   - \( \mu = 104 \),
   - \( \sigma = 24.5 \).

2. **Calculate the z-score**:
   \[
   z = \frac{141.4 - 104}{24.5} = \frac{37.4}{24.5} \approx 1.53
   \]

3. **Find the probability associated with this z-score**:
   We want the probability that an IQ score is greater than 141.4, so we’re looking for \( P(Z > 1.53) \).

   Using the standard normal distribution table, we find:
   \[
   P(Z > 1.53) \approx 1 - 0.9370 = 0.0630
   \]

### Final Answer:
The probability that a randomly selected adult has an IQ greater than 141.4 is approximately **0.0630** or **6.3%**.

Would you like further details or have any questions?

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1. How would the probability change if the mean IQ score increased to 110?
2. What is the probability of an adult having an IQ less than 90?
3. How would we find the probability of an IQ score between two values, such as 100 and 130?
4. What is the impact on probability if the standard deviation changes?
5. How can we interpret probabilities using z-scores?

**Tip:** Understanding how to calculate and interpret z-scores is crucial when working with normal distributions, as it allows you to standardize values and use probability tables effectively.

Find the probability that an adult has an IQ score greater than 141.4, given a normal distribution with mean 104 and standard deviation 24.5. This solution uses z-score calculations and standard normal distribution properties, ideal for high school students learning probability concepts.

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