In a certain distribution of​ numbers, the mean is 100 with a standard deviation of 4. At least what fraction of the numbers are between 88 and 112​?

To solve this problem, we can use **Chebyshev's Inequality**, which applies to any distribution, regardless of its shape.

### Chebyshev's Inequality
Chebyshev's Inequality states that for any number \( k > 1 \), at least \( \frac{1}{k^2} \) of the data values must lie within \( k \) standard deviations of the mean.

### Step-by-Step Solution

1. **Given Information**:
   - Mean \( \mu = 100 \)
   - Standard deviation \( \sigma = 4 \)
   - We need to find the fraction of the numbers that lie between 88 and 112.

2. **Determine the Distance from the Mean**:
   - Lower bound: \( 88 \) 
   - Upper bound: \( 112 \)
   - Distance from the mean to either bound: 
     \[
     100 - 88 = 12 \quad \text{and} \quad 112 - 100 = 12
     \]
   - This distance corresponds to \( k \) standard deviations, so we calculate \( k \) as:
     \[
     k = \frac{12}{4} = 3
     \]

3. **Apply Chebyshev's Inequality**:
   - According to Chebyshev's Inequality:
     \[
     \text{At least} \quad \frac{1}{k^2} \quad \text{of the data lies within} \quad k \quad \text{standard deviations of the mean.}
     \]
   - Substituting \( k = 3 \):
     \[
     \frac{1}{k^2} = \frac{1}{3^2} = \frac{1}{9}
     \]
   - Therefore, at least \( \frac{8}{9} \) (since \( 1 - \frac{1}{9} = \frac{8}{9} \)) of the numbers lie within 3 standard deviations (between 88 and 112).

### Conclusion
At least \(\frac{8}{9}\) of the numbers are between 88 and 112.

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Would you like further details on this solution or have any other questions? 

### Follow-up Questions

1. What is Chebyshev's Inequality used for in statistics?
2. How would this result change if the standard deviation were 5?
3. Can Chebyshev's Inequality be applied to normal distributions?
4. What is the significance of standard deviations in data analysis?
5. How would the calculation differ if the bounds were 90 and 110?
6. What if the distribution were normal? How would you determine the fraction of numbers within a range?
7. How is the mean calculated in a given data set?
8. How can we estimate the standard deviation of a data set?

**Tip:** Chebyshev's Inequality is very useful for distributions that are not normal, as it provides a guaranteed lower bound on the fraction of data within a specified range of the mean.

Explore how Chebyshev's Inequality is applied to solve problems in statistics. Learn the calculation method using a detailed example where the mean is 100 and the standard deviation is 4. Find out what fraction of numbers lie between 88 and 112.

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