Question

Find endpoint(s) on the given normal density curve with the given property. (a) The area to the right of the endpoint on a \(N(50,4)\) curve is about 0.01 (b) The area to the left of the endpoint on a \(N(2,0.05)\) curve is about 0.70 . (c) The symmetric middle area on a \(N(100,20)\) curve is about 0.95 .

Short answer

The endpoints for each part respectively are 40.68; 2.026; and 60.8,139.2.

Step-by-step solution

Understanding of given normal density curves

For each part of the exercise, you'll need to consider the properties of the given normal distribution curve. Properties consist of mean (mu) and standard deviation (sigma). For (a), the curve is \(N(50,4)\) where 50 is the mean and 4 is the standard deviation. For (b), the curve is \(N(2,0.05)\) and for (c), it is \(N(100,20)\).

Calculation for area to the right of the endpoint

Only for (a), use the z-table to find the z-score corresponding to the area of 0.01 which is -2.33. Then, use the z-score formula: z=X-μ/σ, where z is the z-score, X is the score, μ is the mean, and σ is the standard deviation. When you solve for X in the formula for part (a), you'll get \(X = μ + zσ = 50 + (-2.33 * 4) = 40.68\).

Calculation for area to the left of the endpoint

For (b), use z-table to find the z-score corresponding to area of 0.70 which is 0.52. Use the z-score formula again: z=X-μ/σ. Then solve for X and get \(X = μ + zσ = 2 + (0.52 * 0.05) = 2.026\).

Calculation for symmetric middle area

For (c), since the given area is in the middle of the curve, we have to split it into two. That means 0.025 (being 0.5*(1-0.95)) is at each end. The corresponding z-score from the z-table is ±1.96 for the area 0.025. Then, solve for X: \(X = μ + zσ = 100 + (1.96 * 20) = 139.2\). Similarly, for the lower limit: \(X = μ - zσ = 100 - (1.96 * 20) = 60.8\). Therefore, the endpoints are 60.8 and 139.2.