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 square % 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 the standard normal distribution property

The area under the standard normal curve between \(z = - 2\) and \(z = 2\) can be found using the formula \(P(-2<Z<2)=\Phi(2)-\Phi(-2)\), where \(\Phi(z)\) is the cumulative distribution function of the standard normal distribution.
We know that the standard normal distribution is symmetric about \(z = 0\), and \(\Phi(-z)=1 - \Phi(z)\). So \(\Phi(-2)=1-\Phi(2)\)

Step2: Look up the values in the standard - normal table

From the standard - normal table (or using a calculator with a normal - distribution function, e.g., for a TI - 84: normalcdf(-2,2,0,1)), \(\Phi(2)\approx0.9772\), \(\Phi(-2)\approx0.0228\)
Then \(P(-2 < Z < 2)=\Phi(2)-\Phi(-2)=0.9772-(1 - 0.9772)=0.9772-1 + 0.9772=0.9544\)

Answer:

About \(95.44\%\) of the area is between \(z=-2\) and \(z = 2\) (or within 2 standard deviations of the mean).