Questions: Find the value of z such that 0.6826 of the area lies between -z and z. Round your answer to two decimal places.

Solution Steps

To find the value of \( z \) such that 0.6826 of the area lies between \(-z\) and \(z\), we need to use the properties of the standard normal distribution. The area of 0.6826 corresponds to the probability that a standard normal random variable falls within \(-z\) and \(z\). We can use the inverse cumulative distribution function (CDF) of the standard normal distribution to find \( z \).

We need to find the value of \( z \) such that the area under the standard normal distribution curve between \(-z\) and \(z\) is equal to 0.6826. This area represents the probability that a standard normal random variable falls within this range.

Since the area between \(-z\) and \(z\) is 0.6826, we can express this in terms of the cumulative distribution function (CDF) of the standard normal distribution. The probability that a standard normal variable \( Z \) lies between \(-z\) and \(z\) can be written as: \[ P(-z < Z < z) = P(Z < z) - P(Z < -z) = 0.6826 \] Given the symmetry of the normal distribution, we have: \[ P(Z < -z) = 1 - P(Z < z) \] Thus, we can rewrite the equation as: \[ P(Z < z) = \frac{1 + 0.6826}{2} = 0.8413 \]

Using the inverse CDF (also known as the quantile function) for the standard normal distribution, we find: \[ z = \Phi^{-1}(0.8413) \] Calculating this gives us \( z \approx 0.9998 \).

Rounding \( z \) to two decimal places, we obtain: \[ z \approx 1.0 \]

Final Answer