To find the proportion of the normal distribution that lies beyond [tex]\( z = +2.00 \)[/tex], we follow these steps:
1. Understand the Problem:
- We need to determine the area under the standard normal distribution curve to the right of the [tex]\( z \)[/tex]-score [tex]\( +2.00 \)[/tex].
- This area represents the proportion of the distribution in the tail beyond [tex]\( z = +2.00 \)[/tex].
2. Use the Standard Normal Distribution Table:
- The z-score [tex]\( +2.00 \)[/tex] is a measure of how many standard deviations a value is from the mean.
- The cumulative distribution function (CDF) of the standard normal distribution gives the area to the left of a specified z-score.
- So, we first find the area to the left of [tex]\( z = +2.00 \)[/tex] using a standard normal distribution table or CDF.
3. Calculate the Area to the Right:
- Since the total area under the normal distribution curve is 1 (or 100%), the area to the right of [tex]\( z = +2.00 \)[/tex] is calculated by subtracting the CDF value from 1.
- Mathematically, if [tex]\( \Phi(z) \)[/tex] represents the CDF value at [tex]\( z \)[/tex], the area to the right is given by [tex]\( 1 - \Phi(z) \)[/tex].
By looking up the CDF value for [tex]\( z = +2.00 \)[/tex] in a standard normal distribution table, we find that [tex]\( \Phi(2.00) \approx 0.9772 \)[/tex]. Therefore,
[tex]\[
\text{Area to the right of } z = +2.00 = 1 - \Phi(2.00) = 1 - 0.9772 = 0.0228
\][/tex]
By referring to precise computational results, we find that the area to the right of [tex]\( z = +2.00 \)[/tex] is exactly [tex]\( 0.02275013194817921 \)[/tex]. Thus,
[tex]\[
\text{The proportion of the normal distribution in the tail beyond } z = +2.00 \text{ is } 0.02275013194817921 \text{ or approximately } 2.28\%.
\][/tex]
