Question

Determine the two $z-$scores that divide the area under the standard normal curve into a middle $0.90$ area and two outside $0.05$areas.

Short answer

The $z-$score under the typical normal curve with area $0.05$to the left is around $-1.645$. The $z-$score with the area $0.05$to its right under the typical normal curve will be$1.645.$

Step-by-step solution

Given information

The given areas

Middle area is $0.90$

Two outside areas are$0.05.$

Explanation

The given middle area is $0.90$

Two outside areas are $0.05.$

Begin by looking for the area $0.05$in the table's body. It is not in the table, and the closest values are $0.0495$and $0.0505$in the table. $-1.645$is the average of two $z-$scores of $-1.64$and $-1.65$.

As a result, the $z-$score under the typical normal curve with area $0.05$to the left is around $-1.645$. The $z-$score with the area $0.05$to its right under the typical normal curve will be $1.645$using the symmetry property, which can be depicted as

Below is a section of the standard normal table that is relevant: