Question

find the z - score for the given shaded region under the standard normal distribution. round your answer to four decimal places. shaded area = 0.53 z - score = question help: message instructor

Explanation:

Step1: Use the z - score formula

We know that for a standard normal distribution \(Z\sim N(0,1)\), and we are given the area to the left of the \(z\) - score is \(A = 0.53\). We need to find \(z\) such that \(P(Z\leq z)=0.53\).
We can use a standard normal table (or a calculator with a normal - distribution function). The standard normal table gives the cumulative probability \(P(Z\leq z)\) for different values of \(z\).
If we use a calculator (for example, on a TI - 84 Plus: invNorm(0.53,0,1)), where the first argument is the cumulative probability, the second argument is the mean of the standard normal distribution (\(\mu = 0\)), and the third argument is the standard deviation of the standard normal distribution (\(\sigma=1\)).
If we use a standard normal table (z - table), we look for the value closest to \(0.53\) in the body of the table. The standard normal table is symmetric about \(z = 0\), and \(P(Z = 0)=0.5\).

Step2: Calculate the z - score

Using a calculator or software:
invNorm(0.53,0,1)\approx0.0753

Answer:

\(0.0753\)