1. Find
a. �(� < 1.5)
b. �(−1.05 < � < 1.5)
c. �(� > 1.5)

To solve these problems, it looks like you are referring to probabilities involving a standard normal distribution, where \( Z \) represents the standard normal variable (with mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \)).

### a. \( P(Z < 1.5) \)
For a standard normal distribution:
- We need to find the cumulative probability from \( -\infty \) to \( 1.5 \).
- This is the value of the cumulative distribution function (CDF) for \( Z = 1.5 \).

From standard normal distribution tables or using a calculator, the CDF at \( Z = 1.5 \) is:

\[
P(Z < 1.5) = 0.9332
\]

### b. \( P(-1.05 < Z < 1.5) \)
This is the probability that \( Z \) lies between \( -1.05 \) and \( 1.5 \). We calculate this as:

\[
P(-1.05 < Z < 1.5) = P(Z < 1.5) - P(Z < -1.05)
\]

From standard normal tables:
- \( P(Z < 1.5) = 0.9332 \) (as found above).
- \( P(Z < -1.05) \approx 0.1469 \).

Thus:

\[
P(-1.05 < Z < 1.5) = 0.9332 - 0.1469 = 0.7863
\]

### c. \( P(Z > 1.5) \)
The probability of \( Z \) being greater than 1.5 is the complement of \( P(Z < 1.5) \):

\[
P(Z > 1.5) = 1 - P(Z < 1.5)
\]

Using the result from part (a):

\[
P(Z > 1.5) = 1 - 0.9332 = 0.0668
\]

### Final answers:
- a. \( P(Z < 1.5) = 0.9332 \)
- b. \( P(-1.05 < Z < 1.5) = 0.7863 \)
- c. \( P(Z > 1.5) = 0.0668 \)

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Would you like more details or have any questions on how this was solved?

Here are 5 related questions:
1. How do you find the area under the normal curve for other values of \( Z \)?
2. What is the probability \( P(Z < -1.05) \)?
3. How can we calculate \( P(Z > -1.05) \)?
4. What is the standard normal distribution, and why is it useful?
5. How can the normal distribution be applied in real-world problems?

**Tip:** When using standard normal tables, remember that they often provide values for \( P(Z < z) \). To find \( P(Z > z) \), just subtract from 1.

1. Find a. P(Z < 1.5) b. P(-1.05 < Z < 1.5) c. P(Z > 1.5)

Explore how to find probabilities using the standard normal distribution, including P(Z < 1.5), P(-1.05 < Z < 1.5), and P(Z > 1.5). Detailed step-by-step solutions and key concepts in probability and statistics are provided, suitable for students in Grades 10-12.

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