Question

1. for each z - score below, find the percentile (percent of individuals scoring at or below):

a) ( z=-0.47 )
b) ( z = 2.24 )

1. for each z - score below, find the proportion of cases falling above the z:

a) ( z = 0.24 )
b) ( z=-2.07 )

1. for each z - score below, find the area between the mean and the z - score:

a) ( z = 1.17 )
b) ( z=-1.37 )

Explanation:

Problem 2a: z = -0.4

Step 1: Recall z - table usage

To find the percentile (percent of individuals scoring at or below a z - score), we use the standard normal distribution table (z - table). The z - table gives the area to the left of a given z - score.
For \(z=- 0.4\), we look up the value in the z - table. The z - table value for \(z = - 0.4\) (we can think of it as looking at the row for \(-0.4\) and the column for \(0.00\)) is approximately \(0.3446\).

Step 2: Convert to percentile

To convert this area to a percentile, we multiply by \(100\). So the percentile is \(0.3446\times100 = 34.46\%\)

Step 1: Use z - table

We look up the z - score \(z = 2.24\) in the standard normal distribution table. The row for \(2.2\) and the column for \(0.04\) gives us the area to the left of \(z = 2.24\). From the z - table, this area is approximately \(0.9875\).

Step 2: Convert to percentile

Multiply the area by \(100\) to get the percentile. So the percentile is \(0.9875\times100=98.75\%\)

Step 1: Find area to the left

First, we find the area to the left of \(z = 0.24\) using the z - table. The area to the left of \(z = 0.24\) is approximately \(0.5948\).

Step 2: Find area to the right

The proportion of cases falling above the z - score is equal to \(1 -\) the area to the left of the z - score. So we calculate \(1 - 0.5948=0.4052\)

Answer:

\(34.46\%\)

Problem 2b: z = 2.24