Questions: Find the z-scores for which 40% of the distribution's area lies between -z and z. The z-scores are (Use a comma to separate answers as needed. Round to two decimal places as needed.)

Find the z-scores for which 40% of the distribution's area lies between -z and z. The z-scores are (Use a comma to separate answers as needed. Round to two decimal places as needed.)

Transcript text: Find the $z$-scores for which $40 \%$ of the distribution's area lies between $-z$ and $z$. The $z$-scores are $\square$ (Use a comma to separate answers as needed. Round to two decimal places as needed.)

Solution

Solution Steps

Step 1: Understanding the Problem

We need to find the $ z $-scores for which $ 40\% $ of the distribution's area lies between $ -z $ and $ z $. This implies that the cumulative probability from $ -\infty $ to $ -z $ and from $ z $ to $ \infty $ is $ 30\% $ (or $ 0.30 $), leaving $ 70\% $ (or $ 0.70 $) in the middle.

Step 2: Cumulative Probability Calculation

The cumulative probability corresponding to $ z $ is given by: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.70) - \Phi(-\infty) \] Since $ \Phi(-\infty) = 0 $, we have: \[ P = \Phi(0.70) \approx 0.758 \]

Step 3: Finding the $ z $-Scores

The $ z $-scores corresponding to the cumulative probabilities are: \[ z_{start} = -0.70, \quad z_{end} = 0.70 \]