We have been told that the random variable is normally distributed with a mean of μ = 416 and a standard deviation of σ = 55. We want to find the range of values that represent the middle 99.7 % of the distribution.
P(aP(μ-z < X < μ + z) = 99.7 %
Let's recall the Empirical Rule. According to this rule approximately 99.7 % of normally distributed data falls within 3σ from the mean.
P(μ - 3σ < X < μ + 3σ) = 99.7 %
Knowing that μ=416 and σ=55, let's find both ends of the range.
