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### Problem Description:
You have a sample data set of exam scores, with a sample mean of 83.5 and a sample standard deviation of 16. You are asked to determine whether the data comply with Chebyshev's Rule.

### Chebyshev's Rule:
Chebyshev's Rule states that for any dataset, the proportion of values that lie within \(k\) standard deviations of the mean is at least \(1 - \frac{1}{k^2}\), where \(k > 1\).

For instance:
- At least 75% of data lies within 2 standard deviations of the mean.
- At least 89% of data lies within 3 standard deviations of the mean.

### Steps to Solve:
1. **Calculate the interval for \(k = 2\):**
   - The interval within 2 standard deviations from the mean is:
     \[
     \text{Mean} \pm 2 \times \text{Standard Deviation} = 83.5 \pm 2 \times 16 = 83.5 \pm 32 = [51.5, 115.5]
     \]

2. **Count how many data points fall within this interval**:
   - Look at the dataset and count the number of scores within the range \([51.5, 115.5]\).

3. **Determine the proportion**:
   - Calculate the proportion of data points within this range and compare it with 75%.

### Data Set:
- Scores: 24, 56, 68, 68, 69, 78, 80, 82, 83, 84, 86, 88, 90, 91, 91, 92, 92, 92, 92, 93, 96, 97, 97, 100.

### Calculation:
1. **Interval Calculation**:
   - The interval within 2 standard deviations is \([51.5, 115.5]\).

2. **Counting Data Points**:
   - The data points within the interval \([51.5, 115.5]\) are:
     - 56, 68, 68, 69, 78, 80, 82, 83, 84, 86, 88, 90, 91, 91, 92, 92, 92, 92, 93, 96, 97, 97, 100.
   - Total count: 23 out of 24 data points.

3. **Proportion Calculation**:
   - Proportion = \( \frac{23}{24} = 0.9583 \) or 95.83%.
   - Since 95.83% is greater than 75%, the data comply with Chebyshev's Rule for \(k = 2\).

### Conclusion:
- **Answer**: **True**. The data comply with Chebyshev's Rule.

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Would you like any further details or explanations on this problem? Here are five related questions you might consider:

1. What would the proportion be if we considered \(k = 3\) standard deviations?
2. How does Chebyshev's Rule compare with the Empirical Rule?
3. Can Chebyshev's Rule be applied to all distributions, including non-normal distributions?
4. What are the limitations of Chebyshev's Rule in practical scenarios?
5. How would the answer change if the sample standard deviation were different?

**Tip**: Chebyshev's Rule is useful for any dataset, regardless of the distribution, making it more general than rules like the Empirical Rule that assume normality.

Consider the following sample data set of exam scores, where the sample mean is 83.5 and the sample standard deviation is 16. True or False: The data comply with Chebyshev's Rule.

Explore whether a sample data set of exam scores with a mean of 83.5 and a standard deviation of 16 complies with Chebyshev's Rule. Learn about key statistical concepts, including Chebyshev's Theorem and how to calculate the proportion of data points within a specified range. Suitable for students in Grades 10-12.

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