Question

Find endpoint(s) on a \(N(0,1)\) density with the given property. \(\mathbf{P . 1 4 0}(\) a) The area to the right of the endpoint is about 0.02 . (b) The area to the left of the endpoint is about 0.40 .

Short answer

The endpoint for the first condition is about 2.05, and the endpoint for the second condition is about -0.25.

Step-by-step solution

Identify the problem

We are asked to find two endpoints on a standard normal distribution. The standard normal distribution is denoted as \(N(0,1)\), where the mean is 0, and the standard deviation is 1.

Applying the first condition

For the first condition, we need to find an endpoint such that the area (probability) to its right is about 0.02. This is given by \(\mathbf{P . 1 4 0}(Z > z) = 0.02\). Because the total probability under the curve is 1, the probability to the left of the endpoint equals \(1 - 0.02 = 0.98\). Hence, we seek a Z-score \(z\) such that \(\mathbf{P . 1 4 0}(Z < z) = 0.98\). By consulting a standard normal distribution table (or using an online calculator or software tool), we can find that the endpoint \(z\) that meets this condition is about 2.05.

Applying the second condition

For the second condition, we need to find an endpoint such that the area (probability) to its left is about 0.40. This is given by \(\mathbf{P . 1 4 0}(Z < z) = 0.40\). Again, by consulting a standard normal distribution table (or using an online calculator or software tool), we can find that the endpoint \(z\) that meets this condition is about -0.25.