Q. 6.79 - Elementary Statistics

Question

Find the following $z-$scores.

a. $Z_{0.03}$

b. $z_{0.005}$

Step-by-Step Solution

Answer

(a) The$z-$score with the area$0.03$ to its right under the typical normal curve is around $1.88.$

(b)  The $z-$score with a right-hand area of $0.005$ under the typical normal curve is around $2.575.$ 

Part (a) Step 1: Given information

The given area is $0.03.$

Part (a) Step 2: Explanation

The given area is $0.03.$

The $z-$score for which the area under the standard normal curve to its right is $0.03$ will be identical to the z-score for which the area under the standard normal curve to its left is $1-0.03=0.97$.

Begin by looking in the table's body for the area $0.97$. It is not included in the table, and the closest value is $0.9699$. The matching z-score is $1.88$.

As a result, the z-score with the area $0.03$ to its right under the typical normal curve is around $1.88$, as shown in the diagram.

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

Part (b) Step 1: Given information

The given area is $0.005.$

Part (b) Step 2: Explanation

The given area is $0.005$

The $z-$score with 0.005 area under the standard normal curve to its right will be equivalent to the z-score with $1-0.005=0.995$ area to its left.

Begin by looking for the region $0.995$ in the table's body. It is not in the table, and the closest values are $0.9949$ and $0.9951$ in the table. $2.57$ and $2.58$ have an average $z-$score of $2.575$.

As a result, the z-score with a right-hand area of $0.005$ under the typical normal curve is around $2.575$, which can be depicted as

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