Looking up 0.4822 in the standard normal distribution table

Let's clarify the process of finding the \(Z\)-score corresponding to a cumulative probability.

### Step-by-Step Process:

1. **Given:**
   - Cumulative probability \(P(Z < z) = 0.4822\).

2. **Understanding the Problem:**
   - We need to find the \(Z\)-score such that the area to the left of this \(Z\)-score under the standard normal curve is 0.4822.

3. **Using the Standard Normal Table:**
   - The standard normal table (or Z-table) lists cumulative probabilities from the mean (Z = 0) to any positive or negative Z-value. It typically provides the area (probability) to the left of a given Z-value.
   - We look up the cumulative probability 0.4822 in the table to find the corresponding Z-value.

4. **Finding the Z-score:**
   - Upon checking the Z-table for 0.4822, you find that the closest cumulative probability is around 0.4822, which corresponds to a Z-value of approximately \(-0.045\).

### Conclusion:

- Therefore, the Z-score corresponding to a cumulative probability of 0.4822 is approximately \(Z = -0.045\).

This confirms the calculation used earlier to find the value of \(\sigma\) when given \(P(X > 0.69) = 0.5178\).

Would you like further clarification or details?

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Here are 5 questions to expand your understanding:

1. How can the standard normal table be used to find cumulative probabilities for negative Z-scores?
2. What is the relationship between cumulative probability and Z-scores?
3. How would you calculate the Z-score if the table only provided values for positive Z-scores?
4. Can you explain why the Z-score found was negative?
5. How can cumulative probability be used to find the mean or standard deviation in other types of problems?

**Tip:** When using the Z-table, if the exact value isn't found, select the closest value to approximate the Z-score.

Explore how to find the Z-score corresponding to a cumulative probability of 0.4822 in the standard normal distribution. Learn the step-by-step process, use of the standard normal table, and interpretation of cumulative probabilities in statistics. Suitable for advanced high school or college students studying statistics.

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