It appears that you've uploaded an image showing a statistical problem about critical values and confidence intervals. Let me extract and analyze the relevant information to assist you effectively.From the provided image, the task involves calculating critical values for a 90% confidence level with a given alpha ($\alpha = 0.10$).

Steps to solve:

1. Determine alpha and divide by 2:

   • $\alpha = 0.10$
   • Left tail: $\frac{\alpha}{2} = 0.05$
   • Right tail: $1 - \frac{\alpha}{2} = 0.95$

2. Find the Z-scores for these probabilities using a Z-table or statistical tools:

   • For the left tail ($Z_{0.05}$): Find the Z-score for a cumulative probability of 0.05.
   • For the right tail ($Z_{0.95}$): Find the Z-score for a cumulative probability of 0.95.

Critical Values:

• Left-tail critical value: $Z_{0.05} = -1.645$ (negative because it's the left tail).
• Right-tail critical value: $Z_{0.95} = 1.645$.

Final Answer (Rounded to 3 Decimal Places):

• Left critical value: $-1.645$
• Right critical value: $1.645$

Would you like further clarification or help with another part of this question?

5 Related Questions:

1. How do critical values differ for one-tailed vs. two-tailed tests?
2. How do you calculate Z-scores manually without a Z-table?
3. Can this process be used for t-distributions as well? If not, what changes?
4. What are the assumptions behind using Z-scores for confidence intervals?
5. How would the critical values change for a 95% confidence level?

Tip:

Always double-check the rounding and the alpha value to ensure accuracy in confidence interval calculations.