Final grades for all of the sections of the data analysis class (WCOB 1033) for the spring semester are normally distributed with a mean (µ)

Question

Final grades for all of the sections of the data analysis class (WCOB 1033) for the spring semester are normally distributed with a mean (µ) of 75 and a standard deviation (σ) of 13. What is the approximate cutoff value for the top 5% of all the grades?

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Claire 1 month 2021-10-15T14:07:58+00:00 1 Answer 0 views 0

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    2021-10-15T14:09:06+00:00

    Answer:

    The cutoff value for the top 5% of all the grades is 96.385.

    Step-by-step explanation:

    Problems of normally distributed samples are solved using the z-score formula.

    In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

    Z = \frac{X - \mu}{\sigma}

    The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

    In this problem, we have that:

    \mu = 75, \sigma = 13

    What is the approximate cutoff value for the top 5% of all the grades?

    This is the value of X when Z has a pvalue of 0.95. So it is X when Z = 1.645.

    So

    Z = \frac{X - \mu}{\sigma}

    1.645 = \frac{X - 75}{13}

    X - 75 = 13*1.645

    X = 96.385

    The cutoff value for the top 5% of all the grades is 96.385.

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