Question

find the area of the shaded region under the standard normal curve. if convenient, use technology to find the area.
while either technology or the standard normal table can be used to find the area, for this problem, use technology. find the area under the standard normal curve between ( z = - 1.92 ) and ( z = 0 ), rounding to four decimal places.
the area is 0.4726.
therefore, the area of the shaded region is approximately 0.4726.

Explanation:

Step1: Use the property of the standard normal distribution

The standard normal distribution is symmetric about \(z = 0\). The area between \(z=a\) and \(z = 0\) (\(a<0\)) is the same as the area between \(z = 0\) and \(z=-a\). We can use a calculator or statistical software with a normal - distribution function. For example, if using a TI - 84 Plus calculator:
Press 2nd then VARS (DISTR). Select normalcdf(.

Step2: Input the values into the function

The syntax for normalcdf( is normalcdf(lower, upper, \(\mu\), \(\sigma\)). For the standard normal distribution \(\mu = 0\) and \(\sigma=1\). Here, the lower bound \(a=-1.92\), the upper bound \(b = 0\), \(\mu = 0\), \(\sigma = 1\). So we calculate \(P(-1.92<Z<0)=\text{normalcdf}(-1.92,0,0,1)\)

Using a calculator or software, we get the result.

Answer:

The area of the shaded region is approximately \(0.4726\)