Q. 6.69 - Elementary Statistics

Question

In each part, find the area under the standard normal curve that lies between the specified $z$-scores, sketch a standard normal curve and shade the area of interest.

a. $-1$ and $1$

b. $-2$ and $2$

c. $-3$ and $3$

Step-by-Step Solution

Answer

1. The needed surface area will be $0.6826$.
2. The needed surface area will be $0.9544$.
3. The needed surface area will be $0.9974$.

Part(a) Step 1: Given Information

Given in the question that, the points are $-1$ and $1$.

We have to find the area under the standard normal curve that lies between the specified $z$scores, sketch a standard normal curve and shade the area of interest.

Part(a) Step 2: Explanation

The region under the normalized curve lying between $-1$ and $1$ equals the space to the left of $1$ minus the area to the left of $-1$ 

The area to the left of $1$ is $0.8413$, while the area to the left of $-1$ is $0.1587$, according to the standard normal curve.

As a result, the needed surface area will be equal to

$$\begin{gathered} =0.8413-0.1587 \\ =0.6826 \end{gathered}$$

This can be sketched as

Part(b) Step 1: Given Information

Given in the question that, the points are $-2$ and $2$.

Part(b) Step 2: Explanation

The area to the left of $2$ minus the area to the left of $-2$ is the area under the standard normal curve lying between $-2$ and $2$.

The area to the left of $2$ is $0.9772$, while the area to the left of $-2$ is $0.0228$, according to the standard normal curve.

This can be sketched as

Part(c) Step 1: Given Information

Given in the question that, the points are $-3$ and $3$.

Part(c) Step 2: Explanation

The area to the left of $3$ minus the area to the left of $-3$ equals the area under the standard normal curve between $-3$ and $3$.

The area to the left of $3$ is $0.9987$, and the area to the left of $-3$ is $0.0013$, according to the standard normal curve.

This can be sketched as