Questions: True or False: Chebyshev's inequality applies to all distributions regardless of shape, but the empirical rule holds only for distributions that are bell shaped. Choose the correct answer below. A. False, both Chebyshev's inequality and the empirical rule will only work for bell-shaped distributions. B. False, both Chebyshev's inequality and the empirical rule will work for any distribution. C. True, Chebyshev's inequality is less precise than the empirical rule, but will work for any distribution, while the empirical rule only works for bell-shaped distributions. D. False, the empirical rule is less precise than Chebyshev's inequality, but will work for any distribution, while Chebyshev's inequality only works for bell-shaped distributions.

True or False: Chebyshev's inequality applies to all distributions regardless of shape, but the empirical rule holds only for distributions that are bell shaped.

Choose the correct answer below.
A. False, both Chebyshev's inequality and the empirical rule will only work for bell-shaped distributions.
B. False, both Chebyshev's inequality and the empirical rule will work for any distribution.
C. True, Chebyshev's inequality is less precise than the empirical rule, but will work for any distribution, while the empirical rule only works for bell-shaped distributions.
D. False, the empirical rule is less precise than Chebyshev's inequality, but will work for any distribution, while Chebyshev's inequality only works for bell-shaped distributions.

Transcript text: True or False: Chebyshev's inequality applies to all distributions regardless of shape, but the empirical rule holds only for distributions that are bell shaped. Choose the correct answer below. A. False, both Chebyshev's inequality and the empirical rule will only work for bell-shaped distributions. B. False, both Chebyshev's inequality and the empirical rule will work for any distribution. C. True, Chebyshev's inequality is less precise than the empirical rule, but will work for any distribution, while the empirical rule only works for bell-shaped distributions. D. False, the empirical rule is less precise than Chebyshev's inequality, but will work for any distribution, while Chebyshev's inequality only works for bell-shaped distributions.

Solution

Solution Steps

Step 1: Understand Chebyshev's Inequality

Chebyshev's inequality is a statistical rule that applies to any distribution, regardless of its shape. It provides a bound on the probability that a random variable deviates from its mean by more than a certain number of standard deviations. Specifically, for any \( k > 1 \), the probability that a value lies outside \( k \) standard deviations from the mean is at most \( \frac{1}{k^2} \).

Step 2: Understand the Empirical Rule

The empirical rule, also known as the 68-95-99.7 rule, applies specifically to bell-shaped (normal) distributions. It states that:

Approximately 68% of the data lies within 1 standard deviation of the mean.
Approximately 95% of the data lies within 2 standard deviations of the mean.
Approximately 99.7% of the data lies within 3 standard deviations of the mean.

Step 3: Compare the Two Rules

Chebyshev's inequality is more general and applies to any distribution, while the empirical rule is specific to bell-shaped distributions. Chebyshev's inequality is less precise than the empirical rule for bell-shaped distributions but provides a useful bound for distributions of any shape.

Final Answer

The correct answer is C:
\\(\boxed{\text{C. True, Chebyshev's inequality is less precise than the empirical rule, but will work for any distribution, while the empirical rule only works for bell-shaped distributions.}}\\)

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

Next >

Thank you for your feedback.

Copied!