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(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. |

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|(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. |

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|(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. |

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