A random variable is normally distributed with a mean of 25 and a standard deviation of 5. If an observation is randomly selected from the d

Question

A random variable is normally distributed with a mean of 25 and a standard deviation of 5. If an observation is randomly selected from the distribution, what value will be exceeded 85% of the time

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Vivian 6 days 2021-09-13T23:59:43+00:00 1 Answer 0

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  1. Answer:

    Value of 26.93 will be exceeded 85% of the time.

    Step-by-step explanation:

    We are given that a random variable is normally distributed with a mean of 25 and a standard deviation of 5.

    Let X = a random variable

    So, X ~ N(\mu=25,\sigma^{2} =5^{2})

    The z score probability distribution is given by;

                Z = \frac{X-\mu}{\sigma} ~ N(0,1)

    where, \mu = population mean

                \sigma = standard deviation

    Now, we have to find that value which will be exceeded 85% of the time, i.e.;

            P(X > x) = 0.85

        1 – P(X \leq x) = 0.85

             P(X \leq x) = 0.15

             P( \frac{X-\mu}{\sigma} \leq \frac{x-25}{5} ) = 0.15

             P(Z \leq \frac{x-25}{5} ) = 0.15

    Now, in the z table the critical value of x whose less than area is 0.15 is given as 0.3853, i.e;

                     \frac{x-25}{5} = 0.3853

                     x-25=0.3853 \times 5

                        x = 25 + 1.9265 = 26.93

    Therefore, value of 26.93 will be exceeded 85% of the time.

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