Let's go step-by-step to solve each of these problems using the Z-table (Standard Normal Distribution Table).

1. Find the area to the left of 0.75:

To find the area to the left of $Z = 0.75$, we can directly look up the value of 0.75 in the Z-table. The Z-table gives the cumulative area from the far left (negative infinity) up to the given Z value.

• For $Z = 0.75$, the cumulative area is approximately 0.7734.

So, the area to the left of $Z = 0.75$ is 0.7734.

2. Find the area to the right of -0.75:

To find the area to the right of $Z = -0.75$, we can first find the area to the left of $Z = -0.75$ and subtract it from 1.

• From the Z-table, the area to the left of $Z = -0.75$ is approximately 0.2266.

Thus, the area to the right of $Z = -0.75$ is:

$1 - 0.2266 = 0.7734$

So, the area to the right of $Z = -0.75$ is 0.7734.

3. Find the area to the right of 1.96:

The area to the right of $Z = 1.96$ can be found by looking up the area to the left of $Z = 1.96$ in the Z-table and subtracting it from 1.

• From the Z-table, the area to the left of $Z = 1.96$ is approximately 0.9750.

Thus, the area to the right of $Z = 1.96$ is:

$1 - 0.9750 = 0.0250$

So, the area to the right of $Z = 1.96$ is 0.0250.

4. Find the area between -0.32 and -1.19:

To find the area between two Z-values, we calculate the area to the left of each Z value and then subtract the smaller area from the larger area.

• From the Z-table:
  • The area to the left of $Z = -0.32$ is approximately 0.3745.
  • The area to the left of $Z = -1.19$ is approximately 0.1170.

The area between $Z = -0.32$ and $Z = -1.19$ is:

$0.3745 - 0.1170 = 0.2575$

So, the area between $Z = -0.32$ and $Z = -1.19$ is 0.2575.

5. Find the area between 0.5 and 2:

Again, we calculate the area to the left of each Z value and subtract to get the area between the two.

• From the Z-table:
  • The area to the left of $Z = 0.5$ is approximately 0.6915.
  • The area to the left of $Z = 2$ is approximately 0.9772.

The area between $Z = 0.5$ and $Z = 2$ is:

$0.9772 - 0.6915 = 0.2857$

So, the area between $Z = 0.5$ and $Z = 2$ is 0.2857.

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Summary of Answers:

1. Area to the left of 0.75: 0.7734
2. Area to the right of -0.75: 0.7734
3. Area to the right of 1.96: 0.0250
4. Area between -0.32 and -1.19: 0.2575
5. Area between 0.5 and 2: 0.2857

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Would you like more details on how to interpret these areas? Or do you have any questions?

Related Questions:

1. How do you calculate the area for a Z-value that is not listed directly in the Z-table?
2. How does the standard normal distribution apply to real-world problems like in finance or psychology?
3. What are the properties of a standard normal distribution, and why is it symmetric?
4. Can you find the Z-score for a given cumulative probability using the inverse Z-table?
5. What is the significance of the Z-value of 1.96 in hypothesis testing?

Tip:

When using the Z-table, make sure to look at the correct section of the table. The Z-table typically lists values for the Z-score's first two digits (e.g., 0.7 for Z = 0.75) and the second decimal place (0.05 for Z = 0.75). Always double-check your row and column!