Porphyrin is a pigment in blood protoplasm and other body fluids that is significant in body energy and storage. Let x be a random variable

Question

Porphyrin is a pigment in blood protoplasm and other body fluids that is significant in body energy and storage. Let x be a random variable that represents the number of milligrams of porphyrin per deciliter of blood. In healthy circles, x is approximately normally distributed with mean µ = 43 and standard deviation σ = 15. Find the following probabilities. (Round your answers to four decimal places.)

a. x is less than 60
b. x is greater than 16
c. x is between 16 and 60
d. x is more than 60

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MichaelMet 5 years 2021-08-18T05:14:50+00:00 1 Answers 33 views 0

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  1. minhkhue
    0
    2021-08-18T05:15:52+00:00
    Reply

    Answer:

    a) 0.8708 = 87.08% probability that x is less than 60

    b) 0.9641 = 96.41% probability that x is greater than 16.

    c) 0.8349 = 83.49% probability that x is between 16 and 60

    d) 0.1292 = 12.92% probability that x is more than 60.

    Step-by-step explanation:

    When the distribution is normal, we use 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 question, we have that:

    \mu = 43, \sigma = 15

    a. x is less than 60

    This is the pvalue of Z when X = 60. So

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

    Z = \frac{60 - 43}{15}

    Z = 1.13

    Z = 1.13 has a pvalue of 0.8708

    0.8708 = 87.08% probability that x is less than 60

    b. x is greater than 16

    This is 1 subtracted by the pvalue of Z when X = 16. So

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

    Z = \frac{16 - 43}{15}

    Z = -1.8

    Z = -1.8 has a pvalue of 0.0359

    1 – 0.0359 = 0.9641

    0.9641 = 96.41% probability that x is greater than 16.

    c. x is between 16 and 60

    This is the pvalue of Z when X = 60 subtracted by the pvalue of Z when X = 16. We found those in a and b, si:

    0.8708 – 0.0359 = 0.8349

    0.8349 = 83.49% probability that x is between 16 and 60

    d. x is more than 60

    This is 1 subtracted by the pvalue of Z when X = 60.

    So

    1 – 0.8708 = 0.1292

    0.1292 = 12.92% probability that x is more than 60.

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