Scores of an IQ test have a​ bell-shaped distribution with a mean of 100 and a standard deviation of 11. Use the empirical rule to determine the following. ​(a) What percentage of people has an IQ score between 78 and 122? ​(b) What percentage of people has an IQ score less than 78 or greater than 122? ​(c) What percentage of people has an IQ score greater than 111?

Given the IQ test scores with a bell-shaped distribution (normal distribution), a mean of 100, and a standard deviation of 11, we can apply the **Empirical Rule** (68-95-99.7 rule) to answer the questions.

### Empirical Rule:
- **68%** of the data falls within **1 standard deviation** of the mean.
- **95%** of the data falls within **2 standard deviations** of the mean.
- **99.7%** of the data falls within **3 standard deviations** of the mean.

#### (a) What percentage of people has an IQ score between 78 and 122?
- The range 78 to 122 is:
  - \( 78 = 100 - 2 \times 11 \) (2 standard deviations below the mean)
  - \( 122 = 100 + 2 \times 11 \) (2 standard deviations above the mean)
- According to the empirical rule, **95%** of people fall within 2 standard deviations of the mean.
  
So, **95%** of people have an IQ score between 78 and 122.

#### (b) What percentage of people has an IQ score less than 78 or greater than 122?
- Since 95% of the data is between 78 and 122, the remaining percentage (outside this range) is:
  \[
  100\% - 95\% = 5\%
  \]
- This 5% is split evenly on both sides, so:
  - **2.5%** of people have an IQ score less than 78.
  - **2.5%** of people have an IQ score greater than 122.

So, **5%** of people have an IQ score less than 78 or greater than 122.

#### (c) What percentage of people has an IQ score greater than 111?
- \( 111 = 100 + 1 \times 11 \) (1 standard deviation above the mean)
- According to the empirical rule, **68%** of people are within 1 standard deviation of the mean (between 89 and 111).
- This means that \( \frac{100\% - 68\%}{2} = 16\% \) of people have an IQ score greater than 111.

So, **16%** of people have an IQ score greater than 111.

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### 5 Related Questions:
1. What percentage of people has an IQ score between 89 and 122?
2. How would the percentages change if the standard deviation was different?
3. Can we calculate the exact percentages using a z-score instead of the empirical rule?
4. What percentage of people has an IQ score less than 89?
5. What is the probability of having an IQ score between 100 and 122?

### Tip:
The empirical rule is a quick way to estimate probabilities for normal distributions, but for more precise answers, you can use z-scores and a normal distribution table.

Scores of an IQ test have a bell-shaped distribution with a mean of 100 and a standard deviation of 11. Use the empirical rule to determine the following. (a) What percentage of people has an IQ score between 78 and 122? (b) What percentage of people has an IQ score less than 78 or greater than 122? (c) What percentage of people has an IQ score greater than 111?

Learn how to apply the Empirical Rule to an IQ test with a bell-shaped distribution. With a mean of 100 and a standard deviation of 11, find out what percentage of people score between 78 and 122, below 78 or above 122, and greater than 111. Suitable for students in Grades 9-12.

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