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The question asks for the critical value corresponding to a significance level of \(\alpha = 0.20\). Since \(\alpha\) refers to the area in the tails of the distribution, we need to find the positive z-critical value for this level.

For a two-tailed test at \(\alpha = 0.20\), the total area in both tails is 0.20, meaning each tail contains 0.10. The critical value corresponds to the z-score where the area to the left is \(1 - 0.10 = 0.90\).

To find the critical value:

- We look up the z-score that corresponds to a cumulative probability of 0.90. This is approximately **1.28**.

Thus, the positive critical value for \(\alpha = 0.20\) is **1.28**.

Would you like more details on how this is calculated or how to use it in hypothesis testing? 

Here are some follow-up questions that might help expand on this:
1. What does a significance level of \(\alpha = 0.20\) mean in hypothesis testing?
2. How is a z-critical value used in hypothesis testing?
3. What is the difference between a one-tailed and a two-tailed test?
4. How can you calculate a critical value for other significance levels like \(\alpha = 0.05\)?
5. Why is the critical value important in determining the outcome of a statistical test?

**Tip:** In hypothesis testing, always determine whether you're performing a one-tailed or two-tailed test, as it affects the critical value selection.

A colleague wishes to conduct an examination of new cleaning procedures to reduce building cleaning costs. To determine if this is a promising practice, he agrees to collect data and evaluate it using α = 0.20. What critical value should he use? Only list the positive one (Canvas doesn't like listing both!).

Learn how to calculate the positive critical value for a significance level of α = 0.20 in hypothesis testing. This problem involves understanding z-scores, the normal distribution, and the role of critical values in statistical analysis. Suitable for undergraduate students.

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