Chebyshev Theorem: Probability of Parts Within .006 of Mean

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In summary, the standard deviation of diameters of cylindrical parts manufactured on a computer-controlled lathe is .002 millimeter. According to Chebyshev's theorem, the probability that a new part will be within .006 units of the mean for that run is at least 8/9. For the 400 parts made during the run, it can be expected that about 356 of them will lie in that interval.
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Over the range of cylindrical parts manufactured on a computer-controlled lathe, the standard deviation of the diameters is .002 millimeter.

(a) What does Chebyshev's theorem tell us about the probability that a new part will be within .006 units of the mean for that run?

(b) If the 400 parts are made during the run, about what proportion do you expect will lie in the interval in Part a?

I know the answer for (a) the probability is at least 8/9 but I'm not sure about (b). I'm thinking isn't the proportion 8/9 as well, about 356 out of 400 will lie in that interval.
 
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Yes, that is correct.
 

Related to Chebyshev Theorem: Probability of Parts Within .006 of Mean

1. What is Chebyshev's Theorem?

Chebyshev's Theorem is a mathematical rule that helps us understand the probability of a random variable being within a certain distance from its mean. It states that for any data set, no matter the shape of its distribution, a certain proportion of the data will always fall within a given number of standard deviations from the mean.

2. How is Chebyshev's Theorem used in probability?

Chebyshev's Theorem is used to determine the probability of a random variable being within a certain distance from its mean. This can be useful in situations where we do not know the exact shape of the distribution, but we still want to make predictions about the data.

3. What is the formula for Chebyshev's Theorem?

The formula for Chebyshev's Theorem is: P(|X - μ| ≥ kσ) ≤ 1/k², where X is a random variable, μ is the mean, σ is the standard deviation, and k is the number of standard deviations from the mean.

4. How does Chebyshev's Theorem relate to the normal distribution?

Chebyshev's Theorem is a more general version of the Empirical Rule, which is used to describe the normal distribution. The Empirical Rule states that approximately 68% of the data falls within 1 standard deviation from the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. Chebyshev's Theorem can be used for any distribution, not just the normal distribution, and it provides a lower bound for the proportion of data within a given number of standard deviations from the mean.

5. What are some limitations of Chebyshev's Theorem?

One limitation of Chebyshev's Theorem is that it only provides a lower bound for the proportion of data within a given distance from the mean. In some cases, the actual proportion of data within that distance may be much higher. Additionally, the theorem does not take into account the shape of the distribution, so it may not be as accurate for distributions that are not symmetrical.

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