Question

question 3
find z if the standard normal curve area between -z and z is 0.2013.
±0.215
±0.288
±0.315
±0.255

Explanation:

Step1: Recall the property of the standard normal curve

The area between \(-z\) and \(z\) in a standard normal distribution is \(A = 2\Phi(z)- 1\), where \(\Phi(z)\) is the cumulative distribution function of the standard - normal distribution.

We are given that \(A=0.2013\). So, we set up the equation \(2\Phi(z)-1 = 0.2013\).

Step2: Solve the equation for \(\Phi(z)\)

First, add \(1\) to both sides of the equation \(2\Phi(z)-1 = 0.2013\):

\(2\Phi(z)=0.2013 + 1\)

\(2\Phi(z)=1.2013\)

Then divide both sides by \(2\):

\(\Phi(z)=\frac{1.2013}{2}=0.60065\)

Step3: Use the standard - normal table (z - table)

Looking up the value \(0.60065\) in the standard - normal table (z - table).

The standard - normal table gives the \(z\) - value corresponding to a cumulative probability.

We find that \(z\approx0.255\)

Answer:

\(\pm0.255\)