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14 July, 13:39

Assume that X is normally distributed with a mean of 20 and a standard deviation of 2. Determine the following. (a) P (X 24) (b) P (X 18) (c) P (18 X 22) (d) P (14 X 26) (e) P (16 X 20) (f) P (20 X 26)

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  1. 14 July, 16:43
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    a) P (X < 24) = 0.9772

    b) P (X > 18) = 0.8413

    c) P (14 < X < 26) = 0.9973

    d) P (14 < X < 26) = 0.9973

    e) P (16 < X < 20) = 0.4772

    f) P (20 < X < 26) = 0.4987

    Step-by-step explanation:

    Given:

    - Mean of the distribution u = 20

    - standard deviation sigma = 2

    Find:

    a. P (X < 24)

    b. P (X > 18)

    c. P (18 < X < 22)

    d. P (14 < X < 26)

    e. P (16 < X < 20)

    f. P (20 < X < 26)

    Solution:

    - We will declare a random variable X that follows a normal distribution

    X ~ N (20, 2)

    - After defining our variable X follows a normal distribution. We can compute the probabilities as follows:

    a) P (X < 24) ?

    - Compute the Z-score value as follows:

    Z = (24 - 20) / 2 = 2

    - Now use the Z-score tables and look for z = 2:

    P (X < 24) = P (Z < 2) = 0.9772

    b) P (X > 18) ?

    - Compute the Z-score values as follows:

    Z = (18 - 20) / 2 = - 1

    - Now use the Z-score tables and look for Z = - 1:

    P (X > 18) = P (Z > - 1) = 0.8413

    c) P (18 < X < 22) ?

    - Compute the Z-score values as follows:

    Z = (18 - 20) / 2 = - 1

    Z = (22 - 20) / 2 = 1

    - Now use the Z-score tables and look for z = - 1 and z = 1:

    P (18 < X < 22) = P (-1 < Z < 1) = 0.6827

    d) P (14 < X < 26) ?

    - Compute the Z-score values as follows:

    Z = (14 - 20) / 2 = - 3

    Z = (26 - 20) / 2 = 3

    - Now use the Z-score tables and look for z = - 3 and z = 3:

    P (14 < X < 26) = P (-3 < Z < 3) = 0.9973

    e) P (16 < X < 20) ?

    - Compute the Z-score values as follows:

    Z = (16 - 20) / 2 = - 2

    Z = (20 - 20) / 2 = 0

    - Now use the Z-score tables and look for z = - 2 and z = 0:

    P (16 < X < 20) = P (-2 < Z < 0) = 0.4772

    f) P (20 < X < 26) ?

    - Compute the Z-score values as follows:

    Z = (26 - 20) / 2 = 3

    Z = (20 - 20) / 2 = 0

    - Now use the Z-score tables and look for z = 0 and z = 3:

    P (20 < X < 26) = P (0 < Z < 3) = 0.4987
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