The appropriate distribution of the percentages are :
P(μ-2σ≤x≤μ) = 47.5%
P(μ-σ≤x≤μ+σ) = 68%
P(μ-2σ≤x) = 2.5%
P(μ-2σ≤x≤μ-σ) = 13.5%
For a normal distribution :
The mean, μ = 0
The mean, μ = 0The standard deviation, σ = 1
Therefore :
P(μ-2σ≤x≤μ) = P(x≤μ) - P(x≤μ-2σ)
P(x≤μ) - P(x≤μ-2σ)
P(Z ≤ μ) - P(Z ≤ 2(1))
P(Z ≤ 0) - P(Z ≤ 2)
Using the normal distribution table :
0.5 - 0.02275 = 0.47725 = 47.725% (approximately 47.5%)
P(μ-σ≤x≤μ+σ) = P(x≤μ-σ) - P(x≤μ+σ)
P(x≤μ+σ) - P(x≤μ-σ)
P(Z ≤ μ+σ) - P(Z ≤ μ-σ)
P(Z ≤ (0+1) - P(Z ≤ 0-1)
P(Z ≤ 1) - P(Z ≤ - 1)
Using the normal distribution table /
0.84134 - 0.15866 = 0.68268 = 68.268% = (approximately 68%)
P(μ-2σ≤x) = P(μ-2σ)
P(Z ≤ μ-2σ) = P(Z ≤ (0 - 2(1))
P(Z ≤ - 2) = 0.02275 = 2.275% = (approximately 2.5%)
P(μ-2σ≤x≤μ-σ) = P(x≤μ-σ) - P(x≤μ-2σ)
P(x≤μ-σ) - P(x≤μ-2σ)
P(Z ≤ μ-σ) - P(Z ≤ μ-2σ)
P(Z ≤ (0-1) - P(Z ≤ 0-2(1))
P(Z ≤ -1) - P(Z ≤ -2)
From normal distribution table :
0.15866 - 0.02275 = 0.13591 = 13.59% = approximately 13.5%
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