Download (a) When would you use Chebyshev`s theorem and the empirical

(a) When would you use Chebyshev’s theorem and the empirical rule in business? How are they calculated? Provide one real-life
example that requires Chebyshev’s theorem and one that requires the empirical rule.
(b) Why is using Bayes’ theorem important to help answer business-related questions?
(c) What does this theorem allow you to do that traditional statistics do not?
(d) What are some prerequisites for using Bayesian statistics?
(a) (i) Chebyshev's theorem gives an upper bound for the proportion of the data falling within k standard deviations of
the mean of a distribution. The theorem states that, for any population or sample, at least [1 - (1 / k)^2] of the
observations in the data set fall within k standard deviations of the mean, where k  1.
For example we could use Chebyshev's theorem to state the upper limit on the proportion of observations that fall
within, say, 1.5 standard deviation or 2 standard deviations of the mean salary of the employees of an organization. In
this way, we would be able to construct an approximate (rough) confidence interval for the mean salary.
1 - (1/1.5)^2 = 0.5555 → about 55.55% of the salaries are within 1.5 standard deviations of the mean salary
1 - (1/2)^2 = 0.75 → about 75% of the salaries are within 2 standard deviations of the mean salary
(ii) The empirical rule is used for the same purpose as Chebyshev’s theorem, but when we know that the original
observations conform to a normal distribution. The empirical rule states that 68.3%, 95.5%, and 99.7% of all
observations fall within 1, 2, and 3 standard deviations, respectively of the mean.
For example, the control limits of a control chart are drawn 3 standard deviations away from the mean. That means that
99.7% of all observations are expected (or required) to fall within the control limits (assuming the process is in control).
Using such a control chart, we can check if the process is in control or not.
(b) The simplest version of the theorem states that P(A|B) = P(B|A) * P(A) / P(B)
In words, the probability of A given B is equal to the probability of B given A times the probability of A, divided by the
probability of B.
For example, suppose that for a given disease, there is a test that gives accurate positive and negative results 98% of the
time; Also, 3% of the population has the disease.
We can use Baye’s Theorem, to determine the probability that a person has the disease given that the test is positive:
P(has the disease | test is positive) = P(test is positive | has the disease) * P(has the disease) / P(test is positive)
= 0.98 * 0.03 / {0.98 * 0.03 + (1 - 0.98) * (1 - 0.03)}
= 0.6025
Thus, there is a 60.25% chance that given a person tests positive, she actually has the disease.
(c) In Baye’s Theorem, evidence helps confirm our suspicions more if we thought it were more unlikely to occur.
(d) To use Bayesian statistics, one must have some prior knowledge of the opposite probability than one is trying to
determine. For example, if one needs to determine the probability that a person is sick, given the test results, one needs
to have an estimate for the probability that the test comes back positive, given that the person is sick. One also needs to
know the probability of being sick, independent of any test.