Let's tackle this problem step-by-step.

Part a: Finding the Area to the Left of $z = 2.06$

The area to the left of $z = 2.06$ in a standard Normal distribution represents the cumulative probability up to that point. Using a standard Normal distribution table or technology (like a calculator or software), we find the cumulative probability for $z = 2.06$.

1. Find $P(Z \leq 2.06)$:

   • Using a calculator or Normal distribution table, $P(Z \leq 2.06) \approx 0.9803$.

2. Interpretation:

   • This means approximately 98.03% of the area under the standard Normal curve lies to the left of $z = 2.06$.

3. Appropriate Graph (Choice for Part a):

   • The correct graph for this scenario is A, as it shows the area under the curve shaded to the left of $z = 2.06$.

Part b: Finding the Area to the Right of $z = 2.06$

For the area to the right of $z = 2.06$, we subtract the cumulative probability from 1, as the total area under the curve is 1.

1. Find $P(Z > 2.06)$:

   • $P(Z > 2.06) = 1 - P(Z \leq 2.06) = 1 - 0.9803 = 0.0197$.

2. Interpretation:

   • This means that approximately 1.97% of the area under the curve lies to the right of $z = 2.06$.

3. Appropriate Graph (Choice for Part b):

   • Based on the description of the graphs, none specifically seem to describe only the area to the right of $z = 2.06$. However, if there were a graph that showed the right-side shaded, that would be correct.

Summary of Answers:

• Part a: Area to the left of $z = 2.06$ is approximately 0.9803 (Graph A).
• Part b: Area to the right of $z = 2.06$ is approximately 0.0197.

Would you like a sketch of the shaded Normal curve or further details on the calculations?

Follow-up Questions:

1. How would the area change if we wanted to find $P(Z \leq 1.5)$ instead?
2. What does the cumulative probability represent in a Normal distribution?
3. How can we use technology to find probabilities for other values of $z$ in a standard Normal distribution?
4. How would we calculate the area between two points on a Normal distribution, such as between $z = -1$ and $z = 1$?
5. Why is the total area under the standard Normal curve equal to 1?

Tip:

For quick calculations of areas under the Normal curve, many online calculators and statistical software offer functions like norm.cdf (for cumulative distribution) and norm.sf (for survival function or right-tail probability) to help compute these values easily.