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To find the critical value \( z_{\alpha/2} \) for constructing a confidence interval with a confidence level of 87%, follow these steps:

1. **Determine \( \alpha \)**: The confidence level is 87%, which means \( \alpha = 1 - 0.87 = 0.13 \).
   
2. **Find \( \alpha/2 \)**: We divide \( \alpha \) by 2 to account for both tails of the normal distribution. So, \( \alpha/2 = 0.13 / 2 = 0.065 \).

3. **Locate the critical value \( z_{\alpha/2} \)**: The critical value is the z-score that corresponds to a cumulative probability of \( 1 - \alpha/2 = 1 - 0.065 = 0.935 \). Using standard normal distribution tables or a calculator, we find the z-score that gives a cumulative probability of 0.935 is approximately **1.51**.

Thus, the critical value \( z_{\alpha/2} \) for a confidence level of 87% is **1.51** (rounded to two decimal places).

Would you like more details on how to use the z-score table, or do you have any other questions? Here are some related questions:

1. How do we find z-scores for other confidence levels?
2. What is the relationship between confidence level and margin of error?
3. How does the critical value change if we use a t-distribution instead of a normal distribution?
4. How is the width of a confidence interval influenced by sample size?
5. What are some examples of when you would use different confidence levels in practice?

**Tip**: For confidence levels that are commonly used (like 90%, 95%, or 99%), it's helpful to remember their critical values: 1.645 for 90%, 1.96 for 95%, and 2.576 for 99%.

Find the critical value z_{α/2} needed to construct a confidence interval with level 87%.

Learn how to calculate the critical value z_{α/2} for a confidence interval with a confidence level of 87%. This problem involves key statistical concepts, such as the normal distribution and z-scores, and is suitable for high school and college-level students.

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