how is it −49.85% in Suppose that IQ scores have a bell-shaped distribution with a mean of 104104 and a standard deviation of 1515. Using the empirical rule, what percentage of IQ scores are at least 149149?

In the previous step, we determined that we need to subtract from the 50%50% of the data that is above the mean in order to calculate the percentage of data values in the region shaded on the graph to the right.

The empirical rule says that approximately 99.7%99.7% of data lies within 33 standard deviations of the mean. Because of the symmetry of the distribution, half of this 99.7%99.7% is above the mean and half is below. Finally, use this information to calculate what percentage of IQ scores are at least 149149.

The problem you've stated involves understanding the distribution of IQ scores using the empirical rule. However, there appears to be a misunderstanding or typo in the provided explanation that led to the mention of \(-49.85\%\), which isn't a meaningful result in this context. Let’s correctly calculate the percentage of IQ scores that are at least 149 using the empirical rule:

### Given:
- **Mean (\(\mu\))**: 104
- **Standard Deviation (\(\sigma\))**: 15
- **Value of interest (\(x\))**: 149

### Steps:

1. **Find the number of standard deviations (z-score) that 149 is from the mean**:

   \[
   z = \frac{x - \mu}{\sigma} = \frac{149 - 104}{15} = \frac{45}{15} = 3
   \]

   So, 149 is 3 standard deviations above the mean.

2. **Use the empirical rule**:

   - The empirical rule states that approximately 99.7% of data lies within 3 standard deviations of the mean. This means:
     - 68% within 1 standard deviation (\(\mu \pm \sigma\)),
     - 95% within 2 standard deviations (\(\mu \pm 2\sigma\)),
     - 99.7% within 3 standard deviations (\(\mu \pm 3\sigma\)).

3. **Determine the area outside 3 standard deviations**:
   - Since 99.7% of the data lies within 3 standard deviations, \(100\% - 99.7\% = 0.3\%\) lies outside this range.
   - Because the distribution is symmetric, half of this 0.3% is in the lower tail (below \(\mu - 3\sigma\)) and half in the upper tail (above \(\mu + 3\sigma\)).
   - Thus, \(0.3\% \div 2 = 0.15\%\) is above 149.

4. **Calculate the percentage of scores at least 149**:
   - The percentage of IQ scores that are at least 149 is the percentage in the upper tail: **0.15%**.

### Conclusion:

Approximately **0.15%** of IQ scores are at least 149.

### Clarification of the -49.85% Misstatement:
The mention of \(-49.85\%\) seems to be a misunderstanding or a typo. In this problem, we correctly determine that 0.15% of the scores are at least 149. The calculation should not lead to any negative percentages or values. This indicates a possible error in the initial approach or in interpreting the empirical rule's application.

Would you like further details, or do you have any other questions?

Here are 5 related questions that could further expand on this topic:

1. What percentage of IQ scores are below 59 using the empirical rule?
2. If someone has an IQ score of 89, what is their z-score?
3. What is the probability that a randomly chosen person has an IQ score between 89 and 119?
4. How would the empirical rule change if the distribution was not normal but skewed?
5. How many standard deviations away from the mean is an IQ score of 130?

**Tip:** When working with normal distributions, calculating z-scores is a powerful tool for understanding how far a particular value is from the mean. It helps to compare different data points within the same distribution.

Explore how IQ scores are distributed using the empirical rule. Learn how to calculate the percentage of IQ scores above a certain threshold, using a mean of 104 and standard deviation of 15. Suitable for advanced high school students.

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