Question

Suppose a normal distribution has a mean of 120 and a standard deviation of
10. What is P(x >= 100)?

Answer 1

Notice that 100 is 20 less than 120, or equivalently 2 standard deviations σ below the mean µ. So

P(x ≥ 100) = P(x ≥ µ - 2σ)

Recall the empirical rule for normal distributions (these are true for any mean µ and s.d. σ):

• P(µ - σ ≤ x ≤ µ + σ) ≈ 0.65

• P(µ - 2σ ≤ x ≤ µ + 2σ) ≈ 0.95

• P(µ - 3σ ≤ x ≤ µ + 3σ) ≈ 0.997

The probability we want is the same as

P(x ≥ µ - 2σ) = P(µ - 2σ ≤ x ≤ µ + 2σ) + P(µ + 2σ ≤ x)

or

P(x ≥ µ - 2σ) ≈ 0.95 + P(µ + 2σ ≤ x)

which is clearly some number larger than 0.95, so A must be correct.

Answer 2

Suppose a normal distribution has a mean of 120 and a standard deviation, P(x >= 100) is 0.975. Option A. This is further explained below.

What is a normal distribution?

Generally, normal distribution is simply defined as a function that displays the dispersion of numerous unknown parameters as a symmetrical bell-shaped graph.

In conclusion, based on the function

P(x ≥ µ - 2σ) ≈ 0.95 + P(µ + 2σ ≤ x)

the function P(x >= 100) is 0.975

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