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To find the critical value for a significance level of \(\alpha = 0.10\), we must determine the z-score corresponding to the top 10% of the standard normal distribution.

For a two-tailed test, \(\alpha = 0.10\) means that 5% of the distribution is in each tail. However, since the problem specifies to only list the positive critical value, we are interested in the z-score for the upper 10% (one-tailed).

### Step to Determine the Critical Value:

1. **Determine the z-score** corresponding to the upper tail probability of 0.10 (or 90th percentile). Using standard normal distribution tables or a calculator:
   - The critical z-value for \(\alpha = 0.10\) (one-tailed) is **1.28**.

### Answer:
The positive critical value is **1.28**.

Would you like a more detailed explanation or have any questions about this? 

### Related Questions to Explore:
1. What is the critical value for a two-tailed test at \(\alpha = 0.10\)?
2. How does changing the value of \(\alpha\) affect the critical value?
3. What is the difference between one-tailed and two-tailed tests in hypothesis testing?
4. How do you use the critical value in hypothesis testing to make a decision?
5. What is the z-score, and how is it used in statistics?

### Tip:
Always double-check whether your test is one-tailed or two-tailed, as this will affect the critical value you use in your analysis.

The hospital has felt emptier this year for some reason. To determine if this is true, you decide to collect data on the number of empty beds over a period of days and evaluate it using α = 0.10. What critical value should you use? Only list the positive one (Canvas doesn't like listing both!).

Determine the critical value for a one-tailed hypothesis test using α = 0.10. This example involves calculating the Z-score corresponding to the 90th percentile of the standard normal distribution. Learn how to find the critical value in statistics for college-level hypothesis testing.

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