find the indicated area under the curve of the standard normal distribution; then convert it to a percentage…

find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. about ______% of the area is between z = - 2 and z = 2 (or within 2 standard deviations of the mean). about □% of the area is between z = - 2 and z = 2 (or within 2 standard deviations of the mean). (round to two decimal places as needed.)

find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. about ______% of the area is between z = - 2 and z = 2 (or within 2 standard deviations of the mean). about □% of the area is between z = - 2 and z = 2 (or within 2 standard deviations of the mean). (round to two decimal places as needed.)

Answer

Explanation:

Step1: Recall the properties of standard - normal distribution

The cumulative - distribution function of the standard normal distribution $\varPhi(z)$ gives the area to the left of $z$ under the standard - normal curve. The area between $z_1$ and $z_2$ is $A=\varPhi(z_2)-\varPhi(z_1)$.

Step2: Use the standard - normal table

We know that for a standard normal distribution $N(0,1)$, from the standard - normal table, $\varPhi(-2)=0.0228$ and $\varPhi(2)=0.9772$.

Step3: Calculate the area between $z = - 2$ and $z = 2$

$A=\varPhi(2)-\varPhi(-2)=0.9772 - 0.0228=0.9544$.

Step4: Convert to percentage

To convert the area to a percentage, we multiply by 100. So the percentage is $0.9544\times100 = 95.44%$.

Answer:

$95.44$