Let's analyze the problem given in the image.

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.

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.