Questions: Find the value of z such that 0.6318 of the area lies between -z and z. Round your answer to two decimal places.

Transcript text: Find the value of $z$ such that 0.6318 of the area lies between $-z$ and $z$ Round your answer to two decimal places.

Solution

Solution Steps

Step 1: Understand the Problem

We need to find the value of $ z $ such that 0.6318 of the area under the standard normal distribution curve lies between $-z$ and $z$. This means that the cumulative probability from $-z$ to $z$ is 0.6318.

Step 2: Determine the Tail Areas

Since the total area under the standard normal distribution curve is 1, the area outside the interval $[-z, z]$ is $1 - 0.6318 = 0.3682$. This area is split equally between the two tails of the distribution, so each tail has an area of $0.3682 / 2 = 0.1841$.

Step 3: Use the Standard Normal Distribution Table

To find $z$, we need to determine the z-score that corresponds to a cumulative probability of $0.5 + 0.6318/2 = 0.8159$ (since the area from the left tail to $-z$ is 0.1841, and the area from the left tail to $z$ is 0.8159).

Using the standard normal distribution table or a calculator, we find that the z-score corresponding to a cumulative probability of 0.8159 is approximately 0.91.