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The image you shared seems to show a question from an elementary statistics course regarding the normal distribution of IQ scores, which are assumed to be approximately normally distributed with a mean (μ) of 100 and a standard deviation (σ) of 16.

From what I can see:

1. **Question Explanation:**
   The question asks about the proportion of the population that would have an IQ greater than a certain value, likely using the standard deviation rule or the empirical rule.

2. **Empirical Rule (68-95-99.7 Rule):**
   This rule states that:
   - About 68% of data lies within 1 standard deviation from the mean.
   - About 95% lies within 2 standard deviations.
   - About 99.7% lies within 3 standard deviations.

   In your case, the first part of the question mentions values "46 and 154," which correspond to IQ scores within 3 standard deviations from the mean. Since IQ is normally distributed with a mean of 100 and a standard deviation of 16:
   - 3 standard deviations above the mean: \( 100 + (3 \times 16) = 148 \)
   - 3 standard deviations below the mean: \( 100 - (3 \times 16) = 52 \)

   This explains why 46 and 154 are within the range.

3. **Second Question Correction:**
   The second part of the question seems to be calculating a specific probability using the standard normal distribution (Z-scores). You answered 3, but the correct answer appears to be 0.15. 

   This means the correct calculation likely involved finding the area under the normal curve that corresponds to some IQ score. You may have needed to compute the Z-score first and then find the probability using a Z-table.

Would you like help understanding the Z-score calculation or how to derive these probabilities in more detail?

### Related Questions:
1. What is a Z-score, and how is it used to determine probabilities in a normal distribution?
2. How do you calculate the probability that a value falls within a certain number of standard deviations from the mean?
3. Why are the empirical rule percentages (68-95-99.7) so useful in statistics?
4. How can you use a Z-table to find areas under the normal curve?
5. What are common misconceptions when interpreting Z-scores and standard deviations?

#### Tip:
Always verify whether a question asks for a specific Z-score probability or uses the general empirical rule for normal distributions!

The distribution of IQ (Intelligence Quotient) is approximately normal in shape with a mean of 100 and a standard deviation of 16. According to the standard deviation rule, only 3.

This page explains how to apply the Empirical Rule to a normal distribution with a mean of 100 and a standard deviation of 16 to determine probabilities. Learn how Z-scores and the standard deviation rule (68-95-99.7) are used to solve IQ-related statistical problems.

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