Questions: What value of z divides the standard normal distribution so that half the area is on one side and half is on the other? Round your answer to two decimal places.
Transcript text: What value of $z$ divides the standard normal distribution so that half the area is on one side and half is on the other? Round your answer to two decimal places.
Solution
Solution Steps
Step 1: Understanding the Problem
We need to find the value of \( z \) that divides the standard normal distribution such that half the area is on one side and half is on the other. This value corresponds to the median of the standard normal distribution.
Step 2: Calculating the Probability
To determine the value of \( z \), we calculate the probabilities for the ranges from \( -\infty \) to \( 0 \) and from \( 0 \) to \( \infty \).
For the range from \( -\infty \) to \( 0 \):
\[
P = \Phi(0.0) - \Phi(-\infty) = 0.5
\]
For the range from \( 0 \) to \( \infty \):
\[
P = \Phi(\infty) - \Phi(0.0) = 0.5
\]
Both calculations confirm that the area under the curve is equally divided at \( z = 0 \).
Step 3: Conclusion
The value of \( z \) that divides the standard normal distribution into two equal halves is \( 0 \).