Solving using Chebyshev's theorem

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In summary, the thiamine content of a single slice of bread with an average of 0.260 milligrams and standard deviation of 0.005 milligrams must be between 0.220 milligrams and 0.300 milligrams for at least 35/36 of all slices, and between 0.185 milligrams and 0.335 milligrams for at least 80/81 of all slices, according to Chebyshev's theorem.
  • #1
nickar1172
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how do you solve this problem?

Average of one slice contains 0.260 milligrams of vitamin b, with a standard deviation of 0.005 milligrams. According to Chebyshev's theorem, between what values must the thiamine content be of

a) at least 35/36 of all slices of this bread
b)at least 80/81 of all slices of this bread

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is the answer for a) 6 and b) 9?
 
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  • #2
nickar1172 said:
how do you solve this problem?

Average of one slice contains 0.260 milligrams of vitamin b, with a standard deviation of 0.005 milligrams. According to Chebyshev's theorem, between what values must the thiamine content be of

a) at least 35/36 of all slices of this bread
b)at least 80/81 of all slices of this bread

- - - Updated - - -

is the answer for a) 6 and b) 9?

Hi nickar1172! :)

The question ask "between what values", meaning you're supposed to answer each question with a lower bound and an upper bound.

From Chebyshev's theorem $\Pr(|X-\mu|\geq k\sigma) \leq \frac{1}{k^2}$, I do get that $k=6$ for (a) and $k=9$ for (b).

But that's an intermediate result that does not answer the question yet...
 

FAQ: Solving using Chebyshev's theorem

What is Chebyshev's theorem?

Chebyshev's theorem is a mathematical concept that allows us to make statements about the proportion of data values that fall within a certain number of standard deviations from the mean of a data set. It is also known as the Chebyshev inequality or the Bienaymé-Chebyshev inequality.

How is Chebyshev's theorem used in solving problems?

Chebyshev's theorem is used to calculate the minimum proportion of data values that lie within a certain number of standard deviations from the mean of a data set. This can help us identify outliers and evaluate the spread of the data set.

What is the formula for Chebyshev's theorem?

The formula for Chebyshev's theorem is P(|x-μ| < kσ) ≥ 1-1/k², where P is the proportion of data values that fall within k standard deviations from the mean, μ is the mean of the data set, and σ is the standard deviation.

Can Chebyshev's theorem be used for any data set?

Yes, Chebyshev's theorem can be applied to any data set, regardless of the shape or size of the distribution. However, it is most useful for data sets that are approximately bell-shaped or symmetrical.

What are the limitations of Chebyshev's theorem?

Chebyshev's theorem only provides a minimum proportion of data values that fall within a certain number of standard deviations from the mean. It does not tell us the exact proportion or the exact number of data values that fall within that range. Additionally, the theorem assumes that the data set has a finite mean and standard deviation, which may not always be the case.

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