High School

A variable Z is normally distributed with µ=54 and σ=12.3. Find the following probabilities:

a. P(40 < Z ≤ 55.7)

b. P(Z = 64.9)

c. P(Z > 54)

d. P(Z < 48.1)

e. P(Z ≠ 63.4)

f. P(Z ≤ 70)

g. P(Z < 38.2 OR Z > 57.3)

Answer :

To solve these probability problems, we'll use the properties of the normal distribution. A normal distribution is defined by its mean (µ) and standard deviation (σ). For the given variable Z, we have a mean (µ) of 54 and a standard deviation (σ) of 12.3.

To find probabilities for a normally distributed variable, we typically use the standard normal distribution (a normal distribution with a mean of 0 and a standard deviation of 1) and Z-scores. The Z-score formula is:

[tex]Z = \frac{(X - µ)}{σ}[/tex]

Here’s how to approach each part:

a. P(40 < Z \leq 55.7):

  1. Find the Z-score for X = 40:
    [tex]Z_{40} = \frac{(40 - 54)}{12.3} = -1.14[/tex]

  2. Find the Z-score for X = 55.7:
    [tex]Z_{55.7} = \frac{(55.7 - 54)}{12.3} = 0.14[/tex]

  3. Use standard normal distribution tables or calculators to find the probabilities:

    • P(Z < 55.7) corresponds to the cumulative probability for Z = 0.14.
    • P(Z < 40) corresponds to the cumulative probability for Z = -1.14.

  4. Calculate the probability:

    • P(40 < Z \leq 55.7) = P(Z < 55.7) - P(Z < 40)

b. P(Z = 64.9):

In a continuous probability distribution like the normal distribution, the probability of a single point is 0. Therefore,

  • P(Z = 64.9) = 0.

c. P(Z > 54):

Since 54 is the mean, we know the normal distribution is symmetric around the mean.

  • P(Z > 54) = 0.5 because half of the values in a normal distribution fall above the mean.

d. P(Z < 48.1):

  1. Calculate the Z-score for X = 48.1:
    [tex]Z_{48.1} = \frac{(48.1 - 54)}{12.3} = -0.48[/tex]

  2. Find P(Z < -0.48) using a standard normal distribution table or calculator.

e. P(Z ≠ 63.4):

The probability of Z not being a specific value in a continuous distribution is always 1. Therefore,

  • P(Z ≠ 63.4) = 1.

f. P(Z ≤ 70):

  1. Calculate the Z-score for X = 70:
    [tex]Z_{70} = \frac{(70 - 54)}{12.3} = 1.30[/tex]

  2. Find P(Z ≤ 1.30) using a standard normal distribution table or calculator.

g. P(Z < 38.2 OR Z > 57.3):

  1. Calculate the Z-score for X = 38.2:
    [tex]Z_{38.2} = \frac{(38.2 - 54)}{12.3} = -1.28[/tex]

  2. Calculate the Z-score for X = 57.3:
    [tex]Z_{57.3} = \frac{(57.3 - 54)}{12.3} = 0.27[/tex]

  3. Find the cumulative probabilities for these Z-scores:

    • P(Z < 38.2) = P(Z < -1.28)
    • P(Z > 57.3) = 1 - P(Z < 57.3) = 1 - P(Z < 0.27)

  4. Combine for total probability:

    • P(Z < 38.2 OR Z > 57.3) = P(Z < 38.2) + P(Z > 57.3)

Use standard normal distribution tables or calculators for these calculations to get the actual probability values.

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