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!
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
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