To find the area between $Z = 1.37$ and $Z = 2.55$ on the standard normal distribution curve, we will use the Z-table or a standard normal distribution calculator.

Step 1: Find the cumulative probability for $Z = 2.55$

The Z-table gives the cumulative probability (area) to the left of a given Z-score. For $Z = 2.55$, we can find its cumulative probability.

Using a Z-table or standard normal calculator:

• The cumulative probability for $Z = 2.55$ is approximately 0.9946.

Step 2: Find the cumulative probability for $Z = 1.37$

Similarly, for $Z = 1.37$:

• The cumulative probability for $Z = 1.37$ is approximately 0.9147.

Step 3: Calculate the area between $Z = 1.37$ and $Z = 2.55$

The area between these two Z-scores is simply the difference between their cumulative probabilities:

$\text{Area} = P(Z = 2.55) - P(Z = 1.37)$ $\text{Area} = 0.9946 - 0.9147 = 0.0799$

Final Answer:

The area between $Z = 1.37$ and $Z = 2.55$ is approximately 0.0799 or 7.99% of the total area under the standard normal curve.

Would you like more details on how the Z-table works, or any other explanations? Here are some related questions:

1. What is the interpretation of a Z-score in terms of probability?
2. How can you use the standard normal distribution to calculate probabilities for other Z-scores?
3. What does the area under the curve represent in terms of standard normal distribution?
4. How do you calculate probabilities for a different distribution (e.g., binomial)?
5. What is the significance of a Z-score greater than 2 in a normal distribution?

Tip: When calculating areas in a normal distribution, always use the cumulative probability for the Z-scores and subtract to find the area between two points.