To find the percent of the total area under the standard normal curve between the [tex]$z$[/tex]-scores of [tex]$z = -1.2$[/tex] and [tex]$z = 0.8$[/tex], follow these steps:
1. Understand the Problem: We need to find the area under the standard normal distribution curve between two [tex]$z$[/tex]-scores. This area corresponds to the probability that a standard normal variable falls between these two [tex]$z$[/tex]-values.
2. Find CDF Values: First, we determine the cumulative distribution function (CDF) values for these [tex]$z$[/tex]-scores. The CDF of a [tex]$z$[/tex]-score gives the probability that a standard normal variable is less than or equal to that [tex]$z$[/tex]-score.
- For [tex]$z = -1.2$[/tex], the CDF value is approximately [tex]$0.1151$[/tex]. This means that there is an 11.51% probability that a standard normal variable is less than -1.2.
- For [tex]$z = 0.8$[/tex], the CDF value is approximately [tex]$0.7881$[/tex]. This indicates that there is a 78.81% probability that a standard normal variable is less than 0.8.
3. Calculate the Area Between the [tex]$z$[/tex]-scores: To find the area between [tex]$z = -1.2$[/tex] and [tex]$z = 0.8$[/tex], subtract the CDF value at [tex]$z = -1.2$[/tex] from the CDF value at [tex]$z = 0.8$[/tex].
[tex]\[
\text{Area} = \text{CDF}(z = 0.8) - \text{CDF}(z = -1.2)
\][/tex]
[tex]\[
\text{Area} = 0.7881 - 0.1151 = 0.6731
\][/tex]
4. Convert to Percentage: Finally, we convert the area (which is in decimal form) to a percentage by multiplying by 100.
[tex]\[
\text{Percent Area} = 0.6731 \times 100 \approx 67.31\%
\][/tex]
So, the percent of the total area under the standard normal curve between the [tex]$z$[/tex]-scores of [tex]$z = -1.2$[/tex] and [tex]$z = 0.8$[/tex] is approximately [tex]\( 67.31\% \)[/tex].
