Using the Z-Table in Reverse: from the regions to the Z-values

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In a "forward direction" use of the Z-Table, you'll often

  1. Start with a normal distribution with mean \left.\mu\right. and standard deviation \left.\sigma\right.
  2. Transform a value \left.X\right. to a standard normal, to a standard \left.Z\right., via
    Z=\frac{X-\mu}{\sigma}
  3. Find an area value from the Z-Table, corresponding to P(0 \le z \le Z).

In the "backward direction", you do these three steps in reverse:

  1. Starting with an area, you
  2. Find the corresponding \left.Z\right. of the area, considered as P(0 \le z \le Z)
  3. Transform the Z-values from the standard normal to the original normal values, via the transformation

\left.X=\mu+Z\sigma\right.

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