Explanation

This is a confidence interval problem for the population mean when the population variance is unknown. Here are the key steps:

Given:

• Sample size (n) = 30
• Sample mean (x̄) = 4%
• Sample standard deviation (s) = 20%
• Standard error (S_x̄) = 3.651%
• Confidence level = 95%

Key Concepts:

1. Degrees of freedom: Since we're estimating the population standard deviation from the sample, we use n - 1 = 30 - 1 = 29 degrees of freedom

2. Critical t-value: For a 95% confidence interval with 29 degrees of freedom and two-tailed test, we use t_29,2.5 = 2.045

3. Confidence interval formula: $CI = \bar{x} \pm t_{\alpha/2, df} \times S_{\bar{x}}$

Calculation:

• Lower bound = 4% - 2.045 × 3.651% = 4% - 7.466% = -3.466%
• Upper bound = 4% + 2.045 × 3.651% = 4% + 7.466% = 11.466%

Therefore, the 95% confidence interval is [-3.466%, 11.466%].

Why other options are incorrect:

• Option B: Uses incorrect critical value or calculation
• Option C: Uses z-value instead of t-value (likely using 1.96 instead of 2.045)
• Option D: Uses incorrect degrees of freedom or critical value