Q. 6.63 - Elementary Statistics

Question

Determine the area under the standard normal curve that lies between

a. $-2.18 and 1.44$

b. $-2 and -1.5$

c. $0.59 and 1.51$

d. $1.1 and 4.2$

Step-by-Step Solution

Answer

a. The area under the standard normal curve that lies between $-2.18 and 1.44$ is $0.9105$

b. The area under the standard normal curve that lies between $-2 and -1.5$ is $0.0440$

c. The area under the standard normal curve that lies between $0.59 and 1.51$ is $0.2121$

d. The area under the standard normal curve that lies between $1.1 and 4.2$ is $0.1357$

Part (a) Step 1: Given Information

Calculate the area under the standard normal curve that lies between $-2.18 and 1.44$.

Part (a) Step 2: Explanation

Subtracting the smaller area from the bigger area yields the area under the normal curve between two specified $z-$scores.

The area between two $z-$scores is the larger of the two supplied areas minus the smaller of the two given areas.

$0.9251$ is the area under the standard normal curve to the left of $1.44$, while $0.0146$ is the area under the standard normal curve to the left of $-2.18$.

As a result, $0.9251-0.0146=0.9105$ will be the area under the standard normal that lies between $-2.18$and $1.44$.

Part (b) Step 3: Given Information

Calculate the area under the standard normal curve that lies between $-2 and -1.5$ 

Part (b) Step 4: Explanation

A normal curve appears when the smaller area is subtracted from the larger area. 

$Z-$score area is the difference between two supplied areas times the difference between two given areas.

To the left of $-1.2$, there is an area under the standard normal curve of $0.0668$, whereas that of $-1.5$ is $0.0228$.

As a result, the area between $-1.2$ and $-1.5$ will be $0.0668-0.0228=0.0440$under the standard normal.

Part (c) Step 5: Given Information

Calculate the area under the standard normal curve that lies between $0.59 and 1.51$.

Part (c) Step 6: Explanation

By subtracting the smaller area from the larger area, you can find the area under the normal curve between two specified $z-$scores.

$Z-$score areas are equal to the difference between larger and smaller areas between two $z-$scores.

To the left of $1.51$the area under the standard normal curve is $0.9345$.

To the left of $0.59$ the area under the standard normal curve is $0.7236.$

Therefore, $0.9345-0.7224=0.2121$ is the area under the standard normal that lies between $1.51 and 0.59.$

Part (d) Step 7: Given Information

Calculate the area under the standard normal curve that lies between $1.1 and 4.2$  

Part (d) Step 8: Explanation

By subtracting the smaller area from the larger area, you can find the area under the normal curve between two specified $z-$scores.

In a given space, the area between two scores is equal to the larger given space minus the smaller given space.

A standard normal curve to the left of $4.2$has an area of $1.0000,$ and a standard normal curve to the left of $1.1$ has an area of $0.8643$.

In this case, the area under the normal will be $1.0000-0.8643=0.1357$, which lies between $4.2$ and $1.1$.