SOLUTION: Let Z~N(0,1) and z_αthe value such that P(Z≥z_α )=α. Express each of the probabilities below in terms of α. Show the steps used to get to the answers.

Algebra ->  Test -> SOLUTION: Let Z~N(0,1) and z_αthe value such that P(Z≥z_α )=α. Express each of the probabilities below in terms of α. Show the steps used to get to the answers.       Log On


   

Question 1196344: Let Z~N(0,1) and z_αthe value such that P(Z≥z_α )=α. Express each of the probabilities below in terms of α. Show the steps used to get to the answers.
3.1 P(Z
Answer by ElectricPavlov(122) About Me  (Show Source):
You can put this solution on YOUR website!
**1. P(Z >= 0)**
* **Understanding z_α:**
* z_α represents the z-score that corresponds to the right-tail probability of α in a standard normal distribution.
* In other words, P(Z >= z_α) = α
* **Symmetry of Standard Normal Distribution:**
* The standard normal distribution is symmetric around 0.
* Therefore, P(Z >= 0) = 0.5
**2. P(Z <= 5)**
* **Large Z-values:** For very large values of z (like 5), the probability of Z being less than that value is extremely close to 1.
* This is because the standard normal distribution extends indefinitely to the right, but with diminishing probability density.
* **In terms of α:** Since P(Z >= z_α) = α represents a right-tail probability, and P(Z <= 5) covers almost the entire distribution, we can approximate:
* P(Z <= 5) ≈ 1 - α
**In summary:**
* P(Z >= 0) = 0.5
* P(Z <= 5) ≈ 1 - α