Standard Normal Distribution
As different combinations of μ and σ lead to different normal distributions. The Standard normal distribution is defined as the value of normal distribution at μ = 0 and σ = 1. This is known as z-transformation.
Since Z is symmetrical about zero,
- P(Z < -z) = P(Z > z) = 1 – P(Z < z)
- P(Z > -z) = P(Z < z)
Example:
If X ~ (49, 64), calculate:
(i) P (X < 52)
(ii) P (X > 60)
(iii) P (X < 45)
(iv) P (|X – 49| < 5)
Solution:
(i) P (X < 52) = = P(Z < 0.375) = 0.6443
P (X < 52) = 0.6443
(ii) P (X > 60) = \frac{60-49}{\sqrt{64}}) " title="Rendered by QuickLaTeX.com" height="39" width="156" style="vertical-align: -16px;">= P(Z > 1.375) = 1 – P(Z < 1.375) = 0.91466
P (X > 60) = 0.91466
(iii) P(X < 45) = = P(Z < -0.5) = 1 – P(Z < 0.5) = 0.69146
P(X < 45) = 0.69146
(iv) P (|X – 49| < 5) = P ( -5 < (X – 49) < 5)
= P (44 < X < 54)
= P (X < 54) – P (X < 44)
=
= P (Z < 0.625) – P(Z < -0.625)
= P(Z < 0.625) – [ 1 – P(Z < 0.625)]
= 0.73401 – (1 – 0.73401)
P (|X – 49| < 5) = 0.46802
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