Converting Probability to a Z-Score
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Converting Probability to a Z-Score

[From: ] [author: ] [Date: 12-04-12] [Hit: ]
Answer b is correct.First,Second, the area under the Normal curve is perfectly symmetric, so the area from z=0 to z=1 is exactly equal to z=0 to z = -1. If you dont understand this,......
Suppose X is a normally distributed random variable with mean 100 and standard deviation 50 and we want to calculate the probability P(50 ≤ X ≤ 150). Converting the probability to a Z-score we have P(50 ≤ X ≤ 150) =

a) 0
b) 2P(0 ≤ Z ≤ 1)
c) 0.5P(0 ≤ Z ≤ 1)
c) P(0 ≤ Z ≤ 1)

If you could show me the steps to get the answer that would be awesome. Please and thank you in advance!

-
Ruger -

Answer b is correct.

First, you should recognize that P(50 ≤ X ≤ 150) = P(-1 < z < 1) since 50 is exactly 1 standard deviation to the left of 100 and 150 is exactly 1 standard deviation to the right of 100.

Second, the area under the Normal curve is perfectly symmetric, so the area from z=0 to z=1 is exactly equal to z=0 to z = -1. If you don't understand this, pull out a Normal graph and convince yourself that this last statement is true.

Finally, since the areas for these two intervals are exactly equal [-1,0] = [0,1] you state the following:

P(0 ≤ Z ≤ 1) = P(-1 ≤ Z ≤ 0)

Since P(-1 < z < 1) = P(-1 ≤ Z ≤ 0) + P(0 ≤ Z ≤ 1) you can use substitution:

P(-1 < z < 1) = P(0 ≤ Z ≤ 1) + P(0 ≤ Z ≤ 1) = 2P(0 ≤ Z ≤ 1)

Hope that helps
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