Let's go through each part of Questions 5 and 6, using the properties of the standard normal distribution (Z-distribution) and a Z-table to find the probabilities.

Question 5

Given: Standard Normal Distribution, rounded to 3 decimal places.

1. (a) $P(Z > 1.96)$

   • Since $Z$ is greater than a specific value, we need the area to the right of $Z = 1.96$.
   • The Z-table shows that $P(Z < 1.96) = 0.975$.
   • Thus, $P(Z > 1.96) = 1 - 0.975 = 0.025$.

2. (b) $P(Z < 0.8)$

   • We need the area to the left of $Z = 0.8$.
   • The Z-table gives $P(Z < 0.8) = 0.7881$.

3. (c) $P(-0.6 < Z < 1.2)$

   • We calculate $P(Z < 1.2)$ and $P(Z < -0.6)$ and subtract them.
   • From the Z-table:
     • $P(Z < 1.2) = 0.8849$
     • $P(Z < -0.6) = 0.2743$
   • Therefore, $P(-0.6 < Z < 1.2) = 0.8849 - 0.2743 = 0.6106$.

Question 6

Given: Standard Normal Distribution, rounded to at least 4 decimal places.

1. (a) $P(Z > -0.23)$

   • We find $P(Z < -0.23)$ and subtract from 1.
   • The Z-table shows $P(Z < -0.23) = 0.4090$.
   • So, $P(Z > -0.23) = 1 - 0.4090 = 0.5910$.

2. (b) $P(-2.7 < Z < 0.45)$

   • Find $P(Z < 0.45)$ and $P(Z < -2.7)$, then subtract.
   • From the Z-table:
     • $P(Z < 0.45) = 0.6736$
     • $P(Z < -2.7) = 0.0035$
   • Thus, $P(-2.7 < Z < 0.45) = 0.6736 - 0.0035 = 0.6701$.

3. (c) $P(Z < 1.57)$

   • From the Z-table, $P(Z < 1.57) = 0.9418$.

4. (d) $P(Z \geq 2.89)$

   • We find $P(Z < 2.89)$ and subtract from 1.
   • From the Z-table, $P(Z < 2.89) = 0.9981$.
   • So, $P(Z \geq 2.89) = 1 - 0.9981 = 0.0019$.

Summary of Answers

• Question 5:

  • (a) $P(Z > 1.96) = 0.025$
  • (b) $P(Z < 0.8) = 0.788$
  • (c) $P(-0.6 < Z < 1.2) = 0.611$

• Question 6:

  • (a) $P(Z > -0.23) = 0.5910$
  • (b) $P(-2.7 < Z < 0.45) = 0.6701$
  • (c) $P(Z < 1.57) = 0.9418$
  • (d) $P(Z \geq 2.89) = 0.0019$

Would you like a deeper explanation of any step, or do you have any questions?

Related Questions to Explore

1. How would the probabilities change if we used a different Z-score, like $Z > 2.5$?
2. What are the steps to calculate probabilities without a Z-table?
3. How do you interpret a Z-score in real-world contexts?
4. What is the probability of a Z-score falling between two specific values (e.g., $P(1 < Z < 2)$)?
5. Can we find probabilities for non-standard normal distributions?

Tip

When using a Z-table, remember that it typically gives the area to the left of a given Z-score, so you may need to adjust your calculation for right-tail probabilities.