Questions: 15. Given that z is a standard normal random variable, find z for each situation. a. The area to the left of z is .2119. b. The area between -z and z is .9030. c. The area between -z and z is .2052. d. The area to the left of z is .9948. e. The area to the right of z is .6915.

Solution Steps

We are given a standard normal random variable \( z \) and need to find the z-scores for different scenarios involving areas under the standard normal distribution curve. Each part of the problem requires using the properties of the standard normal distribution and the cumulative distribution function (CDF).

For part a, we need to find the z-score such that the cumulative probability to the left of \( z \) is 0.2119. This involves finding the inverse of the CDF for the standard normal distribution.

In part b, the area between \(-z\) and \(z\) is 0.9030. The total area in the tails is \(1 - 0.9030 = 0.0970\). Since the distribution is symmetric, the area in the left tail is \(0.0485\). Therefore, the cumulative probability to the left of \( z \) is \(0.9515\). We find \( z \) using the inverse CDF.

For part c, the area between \(-z\) and \(z\) is 0.2052. The total area in the tails is \(1 - 0.2052 = 0.7948\). The area in the left tail is \(0.3974\). Thus, the cumulative probability to the left of \( z \) is \(0.6026\). We find \( z \) using the inverse CDF.

Final Answer