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.