To solve this problem, we need to find the IQ score, $x$, that corresponds to a given probability value. The question specifies that the IQ scores are normally distributed with:

• Mean ($\mu$) = 100
• Standard deviation ($\sigma$) = 15

The graph shows a probability of 0.2969 to the right of $x$, meaning the area to the right of $x$ in the normal distribution is 0.2969. To find $x$, follow these steps:

Solution:

1. Convert the right-tail probability to a left-tail probability:

   • Since the total area under the normal distribution is 1, the area to the left of $x$ is $1 - 0.2969 = 0.7031$.

2. Find the corresponding z-score:

   • Using a z-table or a statistical calculator, find the z-score that gives a cumulative probability of 0.7031.
   • The z-score for 0.7031 is approximately 0.53.

3. Convert the z-score to the IQ score:

   • Use the z-score formula: $z = \frac{x - \mu}{\sigma}$
   • Rearrange to solve for $x$: $x = z \cdot \sigma + \mu$
   • Substitute the values: $x = 0.53 \cdot 15 + 100$ $x \approx 107.95$
   • Rounding to the nearest whole number, the indicated IQ score is 108.

Answer:

The indicated IQ score is 108.

Would you like further details on any part of this solution?

Here are some related questions for further practice:

1. What would the indicated IQ score be if the area to the right were 0.15?
2. How would the solution change if the mean IQ were 105 instead of 100?
3. What is the probability of an IQ score being greater than 120 in this distribution?
4. How can you find the z-score if only given the left-tail probability?
5. What does a z-score of 0 mean in a normal distribution?

Tip: When solving for an unknown value in a normal distribution, it's often easier to first convert probabilities to z-scores, then use the mean and standard deviation to find the actual data value.