A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time sp

Question

A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be modeled by the following distribution : X~Exp (0.2)
Find the quartile 3. Round to the nearest tenth.

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Latifah 3 months 2021-07-22T00:24:45+00:00 1 Answers 19 views 0

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    2021-07-22T00:26:16+00:00

    Answer:

    The quartile 3 is 0.3

    Step-by-step explanation:

    Exponential distribution:

    The exponential probability distribution, with mean m, is described by the following equation:

    f(x) = \mu e^{-\mu x}

    In which \mu = \frac{1}{m} is the decay parameter.

    The probability that x is lower or equal to a is given by:

    P(X \leq x) = \int\limits^a_0 {f(x)} \, dx

    Which has the following solution:

    P(X \leq x) = 1 - e^{-\mu x}

    The probability of finding a value higher than x is:

    P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^{-\mu x}) = e^{-\mu x}

    X~Exp (0.2)

    This means that m= 0.2, \mu = \frac{1}{0.2} = 5

    Find the quartile 3.

    The 3rd quartile is the 75th percentile, for which P(X \leq x) = 0.75, or P(X > x) = 1 - 0.75 = 0.25

    Since

    P(X > x) = e^{-\mu x}

    e^{-5x} = 0.25

    \ln{e^{-5x}} = \ln{0.25}

    -5x = \ln{0.25}

    x = -\frac{\ln{0.25}}{5}

    x = 0.277

    Rounding to the nearest tenth, the quartile 3 is 0.3.

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