Question

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.)

Explanation:

Step1: Use z - table property

The standard normal distribution is symmetric about z = 0. The area to the left of z = 2 is denoted as $\varPhi(2)$ and the area to the left of z=- 2 is denoted as $\varPhi(-2)$. The area between z=-2 and z = 2 is $A=\varPhi(2)-\varPhi(-2)$.
From the properties of the standard - normal distribution, $\varPhi(-z)=1 - \varPhi(z)$. So, $\varPhi(-2)=1 - \varPhi(2)$. Then $A=\varPhi(2)-(1 - \varPhi(2))=2\varPhi(2)-1$.

Step2: Look up z - values in the z - table

Looking up the value of $\varPhi(2)$ in the standard - normal (z -) table, we find that $\varPhi(2)\approx0.9772$.

Step3: Calculate the area

Substitute $\varPhi(2)\approx0.9772$ into the formula $A = 2\varPhi(2)-1$.
$A=2\times0.9772-1=1.9544 - 1=0.9544$.

Step4: Convert to percentage

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

Answer:

95.44