Questions: Find the z-scores for which 80% 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 80% 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 $80 \%$ 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: Define the Problem

We need to find the $ z $-scores for which $ 80\% $ of the distribution's area lies between $ -z $ and $ z $. This means that the area in the tails of the distribution is $ 20\% $, with $ 10\% $ in each tail.

Step 2: Determine the Cumulative Probability

Since $ 80\% $ of the area is between $ -z $ and $ z $, the cumulative probability for $ z $ is given by: \[ P(-z < X < z) = 0.8 \implies P(X < z) = 0.9 \] Thus, we need to find the $ z $-score corresponding to a cumulative probability of $ 0.9 $.

Step 3: Calculate the Z-Critical Value

Using the formula for the $ z $-critical value: \[ Z = \Phi^{-1}(1 - \frac{\alpha}{2}) \] where $ \alpha = 0.2 $ (the total area in the tails), we find: \[ Z = \Phi^{-1}(1 - 0.1) = \Phi^{-1}(0.9) \]

Step 4: Find the Z-Scores

The calculated $ z $-scores are: \[ z = 1.28 \quad \text{and} \quad -z = -1.28 \]

Step 5: State the Final Result

The $ z $-scores for which $ 80\% $ of the distribution's area lies between $ -z $ and $ z $ are: \[ -1.28, 1.28 \]