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.