Questions: A normal distribution has a mean of 0 and standard deviation of 1. What area is between -2 and 1?

Transcript text: A normal distribution has a mean of 0 and standard deviation of 1. What area is between -2 and 1?

Solution

Solution Steps

Step 1: Define the Problem

We need to find the area under the normal distribution curve between the values \( -2 \) and \( 1 \) for a normal distribution with mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \).

Step 2: Calculate Z-scores

The Z-scores for the given values are calculated as follows:

For the lower bound \( -2 \): \[ Z_{start} = \frac{-2 - 0}{1} = -2.0 \]
For the upper bound \( 1 \): \[ Z_{end} = \frac{1 - 0}{1} = 1.0 \]

Step 3: Calculate the Probability

Using the cumulative distribution function \( \Phi \), we find the probability that a value falls between \( -2 \) and \( 1 \): \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.0) - \Phi(-2.0) \] From the calculations, we have: \[ P \approx 0.8186 \] This means that the probability that a value is between \( -2 \) and \( 1 \) is approximately \( 81.86\% \).