Q. 21 - Elementary Statistics

Question

According to Table II, the area under the standard normal curve that lies to the left of $1.05$ is $0.8531$. Without further reference to Table II, determine the area under the standard normal curve that lies

a. to the right of $1.05$.

b. to the left of $-1.05$.

c. between $-1.05$ and $1.05$

Step-by-Step Solution

Answer

a. A standard normal curve to the right of $1.05$ has an area of $0.1469.$

b. A standard normal curve to the left of $-1.05$ has an area of $0.1469$

c. In the range of $-1.05$ to $1.05$, the area under the normal curve is $0.7062$.

Part (a) Step 1: Given Information

Determine the area under the standard normal curve to the right of $1.05.$

Part (a) Step 2: Explanation

$1.05$has an area under the standard normal curve of $0.8531$.

In calculating the area to the right of any value, the value $1-p$ is multiplied by the value of the area under the standard normal curve to the left of the value.

This area under the standard normal curve is $0.8531$
 to the left of $1.05$.

As a result, the area under the standard normal curve to the right of $1.05$ would be,

$\begin{array}{c}1-p=1-0.8531\\ =0.1469\end{array}$

Below is an illustration of the bell-shaped curve.

Part (b) Step 3: Given Information

Determine the area under the standard normal curve to the right of $-1.05.$

Part (b) Step 4: Explanation

To the left of $1.05$, the area under the standard normal curve is $0.8531.$

Here, $1-p$, where $p$ is the area under the standard normal curve to the left of that value, gives the area under the standard normal curve to the left of any negative number.

The area beneath the typical normal curve to the left of $1.05$ is $0.8531$ in this case.

$0.1469$ is the area under the standard normal curve to the right of $1.05$.

As a result, to the left of $-1.05$, the area under the standard normal curve is $0.1469$.

Below is a diagram of the bell-shaped curve.

Part (c) Step 5: Given Information

Calculate the area under the standard normal curve to between $-1.05\text{ and }1.05$.

Part (c) Step 6: Explanation

To the left of $1.05$, the area under the standard normal curve is $0.8531$.

$p-(1-p)$, where $p$ is the area under the standard normal curve to the left of that value, is the area under the standard normal curve between any negative and its positive value.

The area beneath the typical normal curve to the left of $1.05$ is $0.8531$ in this case.

$0.1469$ is the area under the standard normal curve to the right of $1.05$.

As a result, the following formula is used to determine the area under the standard normal curve between $-1.05$ and $1.05$:

$0.8531-0.1469=0.7062$

Below is a diagram of the bell-shaped curve.