Q. 6.59 - Elementary Statistics

Question

Use Table II to obtain the areas under the standard normal curve required in Exercises 6.59-6.66. Sketch a standard normal curve and shade the area of interest in each problem.

Determine the area under the standard normal curve that lies to the left of

a. $2.24$

b. $-1.56$

c. $0$

d. $-4$

Step-by-Step Solution

Answer

a. The area under the standard normal curve that lies to the left of $2.24$ is $0.9875.$

b. The area under the standard normal curve that lies to the left of $2.24$ is $0.0594$.

c. The area under the standard normal curve that lies to the left of $2.24$ is $0.5000$.

d. The area under the standard normal curve that lies to the left of $2.24$ is $0.0000$.

Part (a) Step 1: Given Information

Use Table II to obtain the areas under the standard normal curve.

Determine the area under the standard normal curve that lies to the left of $2.24$

Part (a) Step 2: Explanation

Because the given number $2.24$ is positive, the conventional normal table of positive $z -$scores is applied. The number obtained by moving down the left hand column labelled $z$ to $2.2$and then across the row to the column labeled $\text{'}0.04\text{'}$ is $0.9875$.

As a result, the area beneath the standard normal to the left of $2.24$ is $0.9875$.

Part (a) Step 3: Given Information

Use Table II to obtain the areas under the standard normal curve.

Determine the area under the standard normal curve that lies to the left of $1.56$.

Part (b) Step 4: Explanation

Because the given number $-1.56$ is negative, the conventional normal table of negative $z$ scores is applied. The number obtained by travelling down the right-hand column labelled $\text{'}z\text{'}$ to $-1.56$and then across the row to the column labelled $0.06$ is $0.0594$.

As a result, the area to the left of $-1.56$ under the standard normal is $0.0594.$

Part (c) Step 5: Given Information

Use Table II to obtain the areas under the standard normal curve.

Determine the area under the standard normal curve that lies to the left of $0.$

Part (c) Step 6: Explanation

The typical table is employed. To begin, reduce the number in the left-hand column labeled $\text{'}z\text{'}$ to $0.0$ and then move across the row to the column labeled $\text{'}0.00 \text{'}.$ The result is $0.5000$.

As a result, the region to the left of $0$ under the usual normal is $0.5000$.

Part (d) Step 7: Given Information

Use Table II to obtain the areas under the standard normal curve.

Determine the area under the standard normal curve that lies to the left of $-4.$

Part (d) Step 8: Explanation

Because the given number $-4.00$ is negative, the conventional normal table of negative $z-$scores is applied. First, move down to $4.0$ in the right-hand column labelled $\text{'}z,\text{'}$ then across the row to the column labelled $\text{'}0.00,\text{'}$ and we will get $0.0000.$

As a result, the region beneath the usual normal to the left of $2.24$ is $0.0000.$