Chebyshev's theorem?

Question

(i) A coffee-maker is regulated so that it takes an average of 5.8 min to brew a cup of coffee with a standard deviation of 0.6 min. According to Chebyshev's theorem, what percentage of the times that this coffee-maker is used will the brewing time take anywhere from 4.6 min to 7 min.
(ii) Give your reason why the Chebyshev's theorem is applied in part (i).
(iii) Do you think that the percentage found in part (i) is accurate? Please explain.

Answer 1

The theorem says that for any random variable X with mean μ and finite variance σ² and k > 0 then

P( | X - μ | < kσ) ≥ 1 - 1/k²

in other words, the probability that X is withing k standard deviations of the mean is no less than 1 - 1/k². this is a very conservative estimate but it works. The proof is easy for continuous random variables and isn't that bad for discrete ones. The most important thing to see here is that it does not matter if the random variable is continuous or discrete, this inequality holds.

(i) The value of 4.6 is 2 standard deviations below the mean and the value 7 is 2 standard deviations above the mean. as such, the percentage of data time the coffee makers will be within this range is 1 - 1/2² = 1 - 1/4 = 3/4 = 75%

(ii) use Chebyshev's theorem because you do not know the distribution of the average brewing time. this theorem is valid for any distribution with finite mean and finite variance.

(iii) it is very conservative. the central limit theorem suggests that if this data was collected from a large enough sample then the average brew time would be normally distributed. if you used the normal distribution you would find 0.9544997 of the data lies within two standard deviations above or below the mean.

Answer 2

Hi,
Chebychev"s Theorem says (to paraphrase) that the fraction of data lying within K standard deviations is at least 1-1/k².
So, your values lie within two standard deviations:
5.8 -2(.6) = 4.6 and
5.8+2(.6) = 7
So, we can say that at least
1-1/2²
=1-1/4
= .75 or 75 % of the data lies within those values.

(ii) The theorem is applied because we don't know the shape of the distribution of the data. If we knew that is approximated a bell-shaped distribution we could use the imperical rule, which says that 95% of the data lies within two standard deviations.

(iii) Notice that the theorem says "at least." That means that more of the data could easily lie within two standard deviations --up to 95% -- depending on how closely the distribution of data approaches a normal curve. So, the number is necessarily subect to some inaccuracy depending on how nearly the distribution of data approaches a normal curve.

Hope this helps.
FE

Answer 3

use chebyshev’s find the least possible fraction of numbers in a data set lying within 5 standard deviation of the means.