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 = -1 and z = 1 (or within 1 standard deviation of the mean).
about □% of the area is between z = -1 and z = 1 (or within 1 standard deviation of the mean).
(round to two decimal places as needed.)

Explanation:

Step1: Use the standard normal distribution table

The standard normal distribution is symmetric about \(z = 0\). The area to the left of \(z=- 1\) is \(P(Z < - 1)\), and the area to the left of \(z = 1\) is \(P(Z < 1)\).
From the standard - normal table, \(P(Z < - 1)=0.1587\) and \(P(Z < 1)=0.8413\).

Step2: Calculate the area between \(z=-1\) and \(z = 1\)

The area between \(z=-1\) and \(z = 1\) is \(P(-1<Z<1)=P(Z < 1)-P(Z < - 1)\).
Substitute the values: \(P(-1 < Z < 1)=0.8413-0.1587 = 0.6826\).

Step3: Convert the area to a percentage

To convert the area (a proportion) to a percentage, multiply by \(100\). So, \(0.6826\times100 = 68.26\%\).

Answer:

\(68.26\)