Suppose X is a normal random variable with mean μ = 100 and standard deviation σ = 7. Find b such that
P(100 ≤ X ≤ b) = 0.3.
HINT [See Example 3.] (Round your answer to one decimal
place.)
b =
Using standard normal conversion,
P(X < x) = P(Z < (x -
) /
)
P(100 < X < b) = 0.3
P(X < b) - P(X < 100) = 0.3
P(Z < ( b - 100) / 7 ) - P(Z < (100 - 100) / 7 ) = 0.3
P(Z < ( b - 100) / 7 ) - P(Z < 0) = 0.3
P(Z < ( b - 100) / 7 ) - 0.5 = 0.3
P(Z < ( b - 100) / 7 ) = 0.8
From Z table, z-score for the probability of 0.8 is 0.8416
( b - 100) / 7 = 0.8416
b = 105.9
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