In a distribution where the mean is 

9090 

and the standard deviation is 

55​, 

find the largest fraction of the numbers that could meet the following requirements.

less than 

6565 

or more than 115115

To solve this problem, we will again use **Chebyshev's Inequality**. This time, we need to determine the largest fraction of numbers that could be either less than 6565 or more than 115115.

### Step-by-Step Solution

1. **Given Information**:
   - Mean \( \mu = 9090 \)
   - Standard deviation \( \sigma = 55 \)
   - We need to find the fraction of the numbers that are less than 6565 or more than 115115.

2. **Determine the Distance from the Mean**:
   - Lower bound: \( 6565 \)
   - Upper bound: \( 115115 \)
   - Distance from the mean to the lower bound: 
     \[
     9090 - 6565 = 2525
     \]
   - Distance from the mean to the upper bound:
     \[
     115115 - 9090 = 106025
     \]
   - Since we need to find the smallest interval around the mean, we use the distance to the **closest bound**, which is 2525. 

   - Calculate \( k \) for this closest bound:
     \[
     k = \frac{2525}{55} \approx 45.91
     \]

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 \( k \approx 45.91 \):
     \[
     \frac{1}{k^2} \approx \frac{1}{(45.91)^2} \approx \frac{1}{2108.6281} \approx 0.0004743
     \]
   - Therefore, at least \( 0.0004743 \) of the data (very small) lies within the interval \( [6565, 115115] \).

   - This implies that the maximum fraction of data outside this interval (either less than 6565 or more than 115115) is approximately \( 1 - 0.0004743 = 0.9995257 \).

### Conclusion
The largest fraction of the numbers that could be less than 6565 or more than 115115 is approximately **0.9995257** or **99.95%**.

---

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

### Follow-up Questions

1. Why is the fraction outside the interval so large?
2. How does Chebyshev's Inequality guarantee the fraction inside the interval?
3. What would happen if the standard deviation were larger?
4. What if the problem asked for numbers between 6565 and 115115 instead?
5. How can Chebyshev's Inequality be applied to asymmetric bounds?
6. Can this method be applied to smaller \( k \) values?
7. How does Chebyshev's Inequality compare to the Empirical Rule for normal distributions?
8. What is the significance of the \( k \) value in determining the distribution of data?

**Tip:** Chebyshev's Inequality is particularly useful when the distribution's shape is unknown, as it provides a bound on how much data can deviate from the mean, regardless of distribution type.

Learn how to apply Chebyshev's Inequality to find the largest fraction of numbers outside the bounds of less than 6565 or more than 115115, given a mean of 9090 and a standard deviation of 55. Explore the step-by-step solution and understand the mathematical reasoning behind the application of Chebyshev's Inequality.

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