Let's solve these probabilities for the standard normal distribution $Z$. I'll calculate each part separately:

Given Information:

The variable $Z$ follows a standard normal distribution, meaning $Z \sim N(0,1)$. We will use the standard normal distribution table (z-table) or calculator to find the probabilities.

Solutions:

(a) $P(Z > 0.58)$

This is the probability that $Z$ is greater than 0.58.

1. Using the z-table or calculator, find $P(Z \leq 0.58)$ and then subtract from 1 to get $P(Z > 0.58)$.
2. $P(Z \leq 0.58) \approx 0.7190$.
3. Therefore, $P(Z > 0.58) = 1 - 0.7190 = 0.2810$.

(b) $P(Z \leq 1.90)$

This is the probability that $Z$ is less than or equal to 1.90.

1. Using the z-table or calculator, find $P(Z \leq 1.90)$.
2. $P(Z \leq 1.90) \approx 0.9713$.

(c) $P(-0.76 < Z < 2.15)$

This is the probability that $Z$ is between -0.76 and 2.15.

1. First, find $P(Z \leq 2.15)$ and $P(Z \leq -0.76)$.
2. $P(Z \leq 2.15) \approx 0.9842$ and $P(Z \leq -0.76) \approx 0.2236$.
3. $P(-0.76 < Z < 2.15) = P(Z \leq 2.15) - P(Z \leq -0.76) = 0.9842 - 0.2236 = 0.7606$.

Answers:

• (a) $P(Z > 0.58) = 0.281$
• (b) $P(Z \leq 1.90) = 0.971$
• (c) $P(-0.76 < Z < 2.15) = 0.761$

Would you like more details on any of these steps, or do you have any questions?

Here are some related questions to explore further:

1. How would you find $P(Z < -1.25)$?
2. What is the probability $P(Z \geq -1.00)$?
3. How would you calculate $P(0 < Z < 1.5)$?
4. What does it mean if $Z \sim N(0,1)$?
5. How would the probabilities change if the distribution was not standard normal?

Tip: When working with standard normal probabilities, it helps to remember that the total area under the curve is 1, so you can often use complements (1 - probability) for calculations.