Questions: If z is a standard normal variable, find the probability P(-0.73<z<2.27). Round to four decimal places. A. 0.4884 B. 1.5400 C. 0.7557 D. 0.2211

Transcript text: If $z$ is a standard normal variable, find the probability $P(-0.73

Solution

Solution Steps

Step 1: Define the Problem

We need to find the probability $ P(-0.73 < z < 2.27) $ where $ z $ is a standard normal variable. This can be expressed using the cumulative distribution function $ \Phi $ of the standard normal distribution.

Step 2: Calculate Z-scores

The Z-scores for the given bounds are:

For the lower bound: $ Z_{start} = -0.73 $
For the upper bound: $ Z_{end} = 2.27 $

Step 3: Apply the Cumulative Distribution Function

The probability can be calculated using the cumulative distribution function: \[ P(-0.73 < z < 2.27) = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(2.27) - \Phi(-0.73) \]

Step 4: Find the Probability

From the calculations, we find: \[ P(-0.73 < z < 2.27) = 0.7557 \]