Question

use the accompanying table of z - scores and percentiles to find the percentage of data items in a normal distribution that lie a. below and b. above a z - score of 2.0
click the icon to view the table of z - scores and percentiles.
a. the percentage of data items that lie below the z - score is □%
(round to two decimal places as needed.)

Explanation:

To solve the problem of finding the percentage of data items below a z - score of 2.0, we can use the properties of the standard normal distribution (z - distribution).

Step 1: Recall the z - table or the empirical rule for the standard normal distribution

The standard normal distribution has a mean of 0 and a standard deviation of 1. For a z - score of \(z = 2.0\), we can use the z - table (or the fact that we know the cumulative probability for a given z - score). The cumulative probability \(P(Z<z)\) gives the proportion of data below the z - score \(z\).

From the standard normal table (z - table), when \(z = 2.00\), the cumulative probability \(P(Z < 2.00)\) can be found as follows:

Looking up \(z=2.0\) in the standard normal table (the table that gives the area to the left of a given z - score), we know that for \(z = 2.00\), the area to the left (the percentage of data below the z - score) is approximately 0.9772.

Step 2: Convert the proportion to a percentage

To convert the proportion 0.9772 to a percentage, we multiply by 100.

\(0.9772\times100=97.72\%\)

Answer:

97.72%