Question

Find endpoint(s) on the given normal density curve with the given property. \(\mathbf{P . 1 4 8}(\) a) The area to the left of the endpoint on a \(N(5,2)\) curve is about 0.10 (b) The area to the right of the endpoint on a \(N(500,25)\) curve is about 0.05

Short answer

The endpoint for the N(5,2) distribution with 0.10 area to the left is approximately 2.436. The endpoint for the N(500,25) distribution with 0.05 area to the right is approximately 541.125.

Step-by-step solution

Calculate the z-score for each scenario

The z-score is defined as the distance from the mean in unit of standard deviation. When the problem gives __the probability 'to the left'__ of the endpoint, we look that percentage up in the z-table or use the percent point function (PPF) or inverse cumulative distribution function from the left. In this case, the z-score for an area of 0.10 to the left endpoint is approximately -1.282. For __'to the right'__ of the endpoint, we calculate as 1 - given probability since the area to the right is equivalent to 1 minus the area to the left. So for the probability 0.05 to the right, we calculate the z-score using 1 - 0.05 = 0.95. The z-score related to this would be approximately 1.645.

Calculate the endpoint for each distribution

We already know that z-score = (X - \(\mu\)) / \(\sigma\), where X is the point in the distribution. We can rearrange this to solve for X, the endpoint we want to find: X = z-score * \(\sigma\) + \(\mu\). For part a, the endpoint X is: -1.282 * 2 + 5 = 2.436. And for part b, the endpoint X is: 1.645 * 25 + 500 = 541.125.