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We have a sample data set of exam scores, with a sample mean of 83.5 and a standard deviation of 16. There are two main questions based on Chebyshev's Rule:

1. Use Chebyshev's Rule to determine the minimum percentage of values that should lie within three standard deviations of the mean.

Chebyshev's Rule states that for any distribution, at least $\frac{1}{k^2}$ of the data will lie within $k$ standard deviations of the mean, where $k$ is any number greater than 1.

For $k = 3$: $\frac{1}{3^2} = \frac{1}{9}$ This gives: $1 - \frac{1}{9} = \frac{8}{9} \approx 0.8889$ So, at least 88.89% of the data should lie within 3 standard deviations of the mean.

2. What percentage of the raw data falls within three standard deviations of the mean?

Now, we need to calculate the range covered by three standard deviations:

• Mean = 83.5
• Standard deviation = 16
• Three standard deviations = $3 \times 16 = 48$

The range within three standard deviations is: $83.5 - 48 = 35.5 \quad \text{to} \quad 83.5 + 48 = 131.5$ Now, we count how many data points fall within this range. The data set is:

$92, 56, 83, 48, 99, 73, 79, 81, 93, 92, 96, 89, 88, 97, 90, 91, 94, 85, 92, 100$

Every data point lies between 35.5 and 131.5, so 100% of the data falls within three standard deviations of the mean.

Thus, the answer to the second question is 100%.

Would you like further clarifications, or more details?

Here are 5 related questions to extend your understanding:

1. How does Chebyshev's Rule differ from the Empirical Rule?
2. Can Chebyshev’s Rule be applied to non-normal distributions?
3. How do we calculate the percentage for $k = 2$ standard deviations using Chebyshev’s Rule?
4. What is the significance of knowing the range of three standard deviations from the mean?
5. How would the calculation change if the sample mean or standard deviation were different?

Tip: Chebyshev’s Rule is versatile and can be used for any distribution, regardless of its shape.