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Find the probability that randomly selected persons IQ is under 85

Find the probability that randomly selected persons IQ is under 85-example-1

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A. The probability of a randomly selected person having an IQ under 85 is approximately 0.15.

B. The probability of a randomly selected person having an IQ over 105 is approximately 0.62.

C. The probability of a randomly selected person having an IQ between 105 and 135 is approximately 0.47.

Since IQ follows a normal distribution with μ = 100 and σ = 15, we can use the standard normal distribution (Z-score) to solve for the probabilities:

a. P(X < 85):

  • First, calculate the Z-score: Z = (85 - 100) / 15 ≈ -1.00.
  • Look up the area to the left of -1.00 in the standard normal distribution table: ~0.1587.
  • Therefore, the probability of a randomly selected person having an IQ under 85 is approximately 15.87%.

b. P(X > 105):

  • Calculate the Z-score: Z = (105 - 100) / 15 ≈ 0.33.
  • Find the area to the right of 0.33 in the table: ~0.6293.
  • This represents the probability of someone's IQ being greater than 105.

c. P(105 < X < 135):

  • Calculate Z-scores for the boundaries: Z1 = (105 - 100) / 15 ≈ 0.33 and Z2 = (135 - 100) / 15 ≈ 2.33.
  • Find the area between Z1 and Z2 in the table: 0.6293 - 0.1587 ≈ 0.4706.
  • This is the probability of an IQ falling within the range of 105 and 135.
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