The scenario you've presented involves testing whether students finish their studies in more than 4 years. Here's how we would calculate the critical values for different tests based on the information provided:

Given Information:

• Sample size (n): 40
• Number of students finishing in more than 4 years (x): 18
• Significance level (α): 10% or 0.10 (since we want to be 90% certain)

Step 1: Determine the Type of Test

The problem asks for different critical values depending on the type of test:

• Two-tailed test: Tests for whether students finish their studies in a time significantly different from 4 years (both more and less than 4 years).
• Right-tailed test: Tests for whether students finish in more than 4 years (which is the direction suggested by the problem).
• Left-tailed test: Tests for whether students finish in less than 4 years.

Step 2: Identify the Z-critical values

For a significance level of 10% (α = 0.10), the critical value depends on the type of test:

• Two-tailed test: This splits α into two tails, so each tail has 5% (0.05).

  • The critical Z-value for a two-tailed test at 90% confidence is ±1.645.

• Right-tailed test: Here, the entire α (10%) is in the right tail.

  • The critical Z-value for a right-tailed test at 90% confidence is 1.28.

• Left-tailed test: Here, the entire α (10%) is in the left tail.

  • The critical Z-value for a left-tailed test at 90% confidence is -1.28.

Critical Values Summary:

• Two-tailed test: Z = ±1.645
• Right-tailed test: Z = 1.28
• Left-tailed test: Z = -1.28

These Z-values are the critical values based on the standard normal distribution for the specified tests.

Let me know if you want more details or have questions! Here are some related questions for deeper understanding:

1. How do we calculate the p-value for a hypothesis test?
2. What is the difference between a one-tailed and a two-tailed test?
3. How does the sample size affect the critical value in hypothesis testing?
4. Why do we use a 90% confidence level, and what would change if we used 95%?
5. What role does the standard error play in hypothesis testing?

Tip: Always check whether your test should be one-tailed or two-tailed based on the research question—it determines how you split the significance level (α).