On a test with a population mean of 75 and standard deviation equal to 16, if the scores are normally distributed, what percentage of scores

Question

On a test with a population mean of 75 and standard deviation equal to 16, if the scores are normally distributed, what percentage of scores fall between 70 and 80?

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Liliana 1 month 2021-10-17T09:19:00+00:00 1 Answer 0 views 0

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    2021-10-17T09:20:38+00:00

    Answer:

    Percentage of scores that fall between 70 and 80 = 24.34%

    Step-by-step explanation:

    We are given a test with a population mean of 75 and standard deviation equal to 16.

    Let X = Percentage of scores

    Since, X ~ N(\mu,\sigma^{2})

    The z probability is given by;

               Z = \frac{X-\mu}{\sigma} ~ N(0,1)    where, \mu = 75  and  \sigma = 16

    So, P(70 < X < 80) = P(X < 80) – P(X <= 70)

    P(X < 80) = P( \frac{X-\mu}{\sigma} < \frac{80-75}{16} ) = P(Z < 0.31) = 0.62172

    P(X <= 70) = P( \frac{X-\mu}{\sigma} < \frac{70-75}{16} ) = P(Z < -0.31) = 1 – P(Z <= 0.31)

                                                  = 1 – 0.62172 = 0.37828

    Therefore, P(70 < X < 80) = 0.62172 – 0.37828 = 0.24344 or 24.34%

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